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Lab Manual

Physics I

LabPaq: PK-1

A Lab Manual of 13 Experiments for Independent Study

Published by Hands-On Labs, Inc.

Physics 1: Lab Manual of Experiments for the Independent Study of Physics

Designed to accompany Physics LabPaq PK-1 081611

LabPaq® is a registered trademark of Hands-On Labs, Inc. (HOL). The LabPaq referenced in this manual is produced by Hands-On Labs, Inc. which holds and reserves all copyrights on the intellectual properties associated with the LabPaq’s unique design, assembly, and learning experiences. The laboratory manual included with a LabPaq is intended for the sole use by that LabPaq’s original purchaser and may not be reused without a LabPaq or by others without the specific written consent of HOL. No portion of any LabPaq manual’s materials may be reproduced, transmitted or distributed to others in any manner, nor may they be downloaded to any public or privately shared systems or servers without the express written consent of HOL. No changes may be made in any LabPaq materials without the express written consent of HOL. HOL has invested years of research and development into these materials, reserves all rights related to them, and retains the right to impose substantial penalties for any misuse.

Author: Peter Jeschofnig, Ph.D. Published by: Hands-On Labs, Inc.

3880 S. Windermere St. Englewood, CO 80110 Phone: 303-679-6252 Toll-free: 1-866-206-0773 Fax: 270-738-0979

www.LabPaq.com

E-mail: [email protected]

Printed and bound in the United States of America.

ISBN: 978-1-866151-39-0

The experiments in this manual have been and may be conducted in a regular formal laboratory or classroom setting with the user providing their own equipment and supplies. The manual was especially written, however, for the benefit of independent study students who do not have convenient access to such facilities. It allows them to perform physics experiments at home or elsewhere by using LabPaq PK-1, a collection of experimental equipment and supplies specifically packaged by Hands-On Labs, Inc. to accompany this manual.

Use of this manual and authorization to perform any of its experiments are expressly conditioned upon the user reading, understanding, and agreeing to abide by all the safety precautions contained herein. Although the author and publisher have exhaustively researched all sources to ensure the accuracy and completeness of the information contained in this book, we assume no responsibility for errors, inaccuracies, omissions or any other inconsistency herein. Any slight of people, organizations, materials or products is unintentional.

Table of Contents Introduction .................................................................................................................................. 4

Important Information to Help Students Study Science ..................................................... 4 WELCOME TO THE WORLD OF SCIENCE! ................................................................................ 4

Laboratory Equipment and Techniques ........................................................................... 13 Use, Disposal, and Cleaning Instructions for Common Materials ................................... 19

HOW TO WRITE LAB NOTES AND LAB REPORTS .................................................................. 21 Lab Notes .......................................................................................................................... 21 Lab Reports ....................................................................................................................... 23 Laboratory Drawings ......................................................................................................... 27 Visual Presentation of Data .............................................................................................. 28 Computer Graphing Using MS Excel ................................................................................. 32

SAFETY CONCERNS ............................................................................................................... 40 Basic Safety Guidelines .................................................................................................... 41 Material Safety Data Sheets ............................................................................................. 46 Science Lab Safety Reinforcement Agreement ............................................................... 50

EXPERIMENTS 1. Experimental Errors and Uncertainty .............................................................................. 53 2. Measurement: Length, Mass, Volume, Density, and Time ............................................ 63 3. Trigonometric Measurements ......................................................................................... 82 4. Data Collection ................................................................................................................. 90 5. Acceleration ................................................................................................................... 104 6. Friction............................................................................................................................ 112 7. Simple Machine – Lever ................................................................................................ 122 8. Simple Machine – Pulleys ............................................................................................. 132 9. Pendulum and the Calculation of g ............................................................................... 138 10.Centripetal Acceleration ................................................................................................ 145 11.Hooke’s Law .................................................................................................................. 161 12.Specific Heat Capacity of Metals .................................................................................. 169 13.Determining the Speed of Sound ................................................................................. 177

APPENDIX

Using Statistics .................................................................................................................... 192

Introduction Important Information to Help Students Study Science

Version 09.3.05

WELCOME TO THE WORLD OF SCIENCE! Don't be afraid to take science courses. When you complete them, you will be very proud of yourself and will wonder why you were ever afraid of the “S” word – Science! After their first science course most students say they thoroughly enjoyed it, learned a lot of useful information relevant to their personal lives and careers, and only regret not having studied science sooner. Science is not some mystery subject comprehended only by eggheads. Science is simply a way of learning about our natural world and how it works by testing ideas and making observations. Learning about the characteristics of the natural world, how those characteristics change, and how those characteristics interact with each other make it easier to understand ourselves and our physical environment and to make the multitude of personal and global decisions that affect our lives and our planet. Plus, science credits on an academic transcript are impressive, and your science knowledge may create some unique job opportunities. All sciences revolve around the study of natural phenomena and require hands-on physical laboratory experiences to permit and encourage personal observations, discovery, creativity, and genuine learning. As increasing numbers of students embrace online and independent study courses, laboratory experiences must remain an integral part of science education. This lab manual’s author and publisher are science educators who welcome electronic technology as an effective tool to expand and enhance instruction. However, technology can neither duplicate nor replace learning experiences afforded to students through traditional hands-on laboratory and field activities. This does not mean that some experiments cannot or should not be replaced or reinforced by computer simulations; but any course of science study must also provide sufficient hands-on laboratory and field experiences to:

 Engage students in open-ended, investigative processes by using scientific problem solving.

 Provide application of concepts students have seen in their study materials which

reinforce and clarify scientific principles and concepts.

 Involve multiple senses in three-dimensional rather than two-dimensional learning experiences that are important for greater retention of concepts and for accommodation of different leaning styles.

 Stimulate students to understand the nature of science including its unpredictability

and complexity.

 Provide opportunities to engage in collaborative work and to model scientific attitudes and behavior.

 Develop mastery of techniques and skills needed for potential science, engineering, and technology careers.

 Ensure advanced placement science courses transfer to college credit.

The knowledge gained from science courses with strong laboratory components enables students to understand in practical and concrete ways their own physical makeup, the functioning of the natural world around them, and contemporary scientific and environmental issues. It is only by maintaining hands-on laboratory experiences in our curricula that the brightest and most promising students will be stimulated to learn scientific concepts and avoid being turned-off by lecture- and textbook-only approaches. Physical experimentation may offer some students their only opportunity to experience a science laboratory environment. All students – as potential voters, parents, teachers, leaders, and informed citizens – will benefit from a well-rounded education that includes science laboratory experiences, when it is time for them to make sound decisions affecting the future of their country and the world.

19th century scientist, Ira Remsen (1846-1927) on the subject of Experimentation:

This lab manual can be used by all students, regardless of the laboratory facilities available to them. The experiments are based on the principles of micro- and small-scale science which have been successfully used in campus laboratories for decades. LabPaq’s micro- and small-scale experiments can also be performed at home, in a dorm room, or at a small learning center that lacks a formal laboratory. What are Micro- and Small-scale Experiments? You may be among the growing number of students to take a full-credit, laboratory science course through independent study, due to the development and perfection of micro-scale and small-scale experimentation techniques over the past half century. While experimentation on any scale is foundational to fully understanding science concepts, science courses in the past have required experimentation to be performed in the campus laboratory due to the potential hazards inherent in traditional experimentation. Potential hazards, increasing chemical, specimen, and science equipment costs, and environmental concerns made high schools, colleges, and universities reexamine the traditional laboratory methods used to teach science. Scientists began to scale down the quantities of materials and the size of equipment used in experiments and found reaction results remained unchanged. Over time, more and more traditional science experiments were redesigned to be performed on micro and small scales. Educational institutions eventually recognized that the scientific reaction, not the size of the reaction, facilitates learning. Successive comparative assessments have proven that students’ learning is not impaired by studying small-sized reactions. Many assessments even suggest that science learning is enhanced by small-scale experimentation. The primary pioneer and most prominent contributor to micro- and small-scale experimentation was Dr. Hubert Alyea, a chemistry professor at Princeton University, who began utilizing micro-scale experiments in the 1950s. Dr. Alyea reformatted numerous chemistry experiments and also designed many of the techniques and equipment used in micro- and small-scale science today. In the mid-1990s, Dr. Peter Jeschofnig of Colorado Mountain College pioneered the development of LabPaq’s academically aligned, small-scale experiments that can be performed at home. Hands-On Labs, Inc. has subsequently proven that students can actually perform LabPaq's rigorous science experiments at home and still achieve an equivalent, if not higher, level of learning than their campus-based peers.

The Organization of this Lab Manual Before proceeding with your experiments, please thoroughly read and understand each section of this lab manual, so you understand what is expected of you.

Introduction and How to Study Science: These sections include important information about general scientific subject matter and specific information about effectively studying science and conducting science experiments. Read these sections carefully and take them to heart!

How to Perform an Experiment and Laboratory Equipment and Techniques: Adhering to the procedures described in these sections will greatly facilitate experimental activities. The laboratory techniques and equipment described primarily apply to full-scale experiments and formal laboratories; however, knowledge of these items is important to a basic understanding of science and is relevant to home-based experimentation.

How to Write Lab Notes and Lab Reports: Like all serious scientists, you must record formal notes detailing your activities, observations, and findings for each experiment. These notes will reinforce your learning experiences and science knowledge and provide the basis from which you will prepare Lab Reports for your instructor. This section explains how these documents should be organized and prepared.

Safety Concerns: The Basic Safety Guidelines and Safety Reinforcement Agreement are the most important sections of this lab manual and should be reviewed before each experiment. The safety sections are relevant to both laboratory and non-laboratory experimentation. The guidelines describe potential hazards as well as basic safety equipment and safety procedures designed to avoid such hazards.

Required Equipment and Supplies: If you are performing these experiments in a non- laboratory setting, you must obtain the LabPaq specifically designed to accompany this lab manual. The LabPaq includes all the basic equipment and supplies needed to complete the experiments, except for minor items usually found in the average home or obtained at local stores. At the beginning of each experiment you will find a materials section listing which items are found in the LabPaq and which items you will need to provide. Review this list carefully before you begin an experiment to ensure you have all required items.

Experiments: The experiments included in this lab manual were specifically selected to accompany related course materials for a traditional academic term. These experiments emphasize a hands-on, experimental approach for gaining a sound understanding of scientific principles. The lab manual’s rigorous Lab Report requirements help reinforce and communicate your understanding of each experiment’s related science principles and strengthen your communication skills. This traditional, scientific method approach to learning science reflects the teaching philosophy of the authors, Hands-On Labs, Inc., and science educators around the globe.

HOW TO STUDY SCIENCE It is unfortunate that many people develop a fear of science somewhere early in life. Yes, the natural sciences are not the easiest subjects to learn; but neither are they the hardest. Like in any other academic endeavor, if you responsibly apply yourself, conscientiously study your course materials, and thoughtfully complete your assignments, you will learn the material. Following are some hints for effectively studying science and any other subject, both on or off campus. Plan to Study: You must schedule a specific time and establish a specific place in which to seriously devote yourself to your studies. Think of studying like you would think of a job. Jobs have specific times and places in which to get the work done, and studying should be no different. Just as television, friends, and other distractions are not permitted on a job, they should not be permitted to interfere with your studies. If you want to do something well, you must be serious about it, and you cannot learn when you are distracted. Only after you have finished your studies should you allow time for distractions. Get in the Right Frame of Mind: Think positively about yourself and what you are doing. Put yourself in a positive frame of mind to enjoy what you are about to learn, and then get to work. Organize any materials and equipment you will need in advance so you don't have to interrupt your work later. Read your syllabus and any other instructions and know exactly what your assignment is and what is expected of you. Mentally review what you have already learned. Write down any questions you have, and then review previous materials to answer those questions. Move on, if you haven't found the answer after a reasonable amount of time and effort. The question will germinate inside your mind, and the answer will probably present itself as you continue your studies. If not, discuss the question later with your instructor. Be Active with the Material: Learning is reinforced by relevant activity. When studying, feel free to talk to yourself, scribble notes, draw pictures, pace out a problem, or tap out a formula. The more physically active things you do with your study materials, the better you will learn. Have highlighters, pencils, and note pads handy. Highlight important data, read it out loud, and make notes. If there is a concept you are having problems with, stand up and pace while you think it through. Try to see the action taking place in your mind. Throughout your day, try to recall things you have recently learned, incorporate them into your conversations, and teach them to friends. These activities will help to imprint the related information in your brain and move you from simple knowledge to true understanding of the subject matter.

Do the Work and Think about What You Are Doing: Sure, there are times when you might get away with taking a shortcut in your studies, but in doing so you will probably shortchange yourself. The things we really learn are the things we discover ourselves, which is why we don't learn as much from simple lectures, passive videos, or someone simply telling us the answers to our questions. Discovery learning – figuring things out for ourselves – is the most effective and long-lasting form of learning. When you have an assignment, don't just go through the motions. Enjoy your work, think about what you are doing, be curious, ask yourself questions, examine your results, and consider the implications of your findings. These critical thinking techniques will improve and enrich your learning process. When you complete your assignments independently and thoroughly, you will be genuinely knowledgeable and can be very proud of yourself. How to Study Independently There is no denying that learning through any method of independent study is very different from learning through classes held in traditional classrooms. It takes a great deal of personal motivation and discipline to succeed in a course of independent study where there are no instructors or fellow students to give you structure and feedback. These problems are not insurmountable, and meeting the challenges of independent study can provide tremendous personal satisfaction. The key to successful independent study is having a personal study plan and the personal discipline to stick to that plan. Properly Use Your Learning Tools: The basic tools for web courses and other distance learning methods are often similar, consisting of computer software, videos, textbooks, and study guides. Check with your course instructor to make sure you acquire all the materials you will need. You can obtain these items from campus bookstores, libraries, or the Internet. Related course lectures and videos may even be broadcast on your local public and educational television channels. If you choose to do your laboratory experimentation independently, you will need the special equipment and supplies described in this lab manual and contained in its companion LabPaq. For each study session, first work through the appropriate sections of your course materials, because these serve as a substitute for classroom lectures and demonstrations. Take notes as you would in a regular classroom. Actively work with any computer and text materials, carefully review your study guide, and complete all related assignments. If you do not feel confident about the material covered, repeat the previous steps until you do. It is wise to always review your previous work before proceeding to a new section to reinforce what you’ve previously learned and prepare you to better absorb new information. Actual experimenting is among the last things done in a laboratory session.

Plan to Study: A normal science course with a laboratory component may require you to spend as many as 15 hours a week studying and completing your assignments. To really learn new material requires at least three hours of study time each week for each hour of course credit taken. This applies as equally to independent study as it does to regular classroom courses. On a school campus science students are usually in class for three hours and in the laboratory for two to three hours each week. Then, they still need at least nine hours to read their text and complete their assignments. Knowing approximately how much time is required will help you formulate a study plan at the beginning of the course. Schedule Your Time Wisely: The more often you interact with study materials and call them to mind, the more likely you are to reinforce and retain the information. It is much better to study in several short blocks of time rather than in one long, mind-numbing session. Accordingly, you should schedule several study periods throughout the week or during each day. Please do not try to do all of your study work on the weekends! You will burn yourself out, you won't learn as much, and you will probably end up feeling miserable about yourself and science too. Wise scheduling can prevent such unpleasantness and frustration. Choose the Right Place for Your Home Laboratory: The best place to perform at-home experiments will be determined by the nature of the individual experiments. However, this place is usually an uncluttered room where a door can be closed to keep out children and pets; a window or door can be opened for fresh air ventilation and fume exhaust; there is a source of running water for fire suppression and cleanup; and there is a counter or tabletop work surface. A kitchen usually meets all these requirements. Sometimes the bathroom works too, but it can be cramped and subject to interruptions. Review each experiment before starting any work to help you select the most appropriate work area. Because some of the equipment and supplies in your LabPaq may pose dangers to small children and animals, always keep safety in mind when selecting a work area, and always choose an area where you cannot be disturbed by children or pets. Use a Lab Partner: While the experiments in the LabPaq can be performed independently, it is often fun and useful to have a lab partner to discuss ideas with, help take measurements, and reinforce your learning process. Whether your partner is a parent, spouse, sibling, or friend, you will have to explain what you are doing, and in the process of teaching another, you will better teach yourself. Always review your experiments several days ahead of time so you have time to line up a partner if needed. Perform Internet Research: Students in today’s electronic information age are often unaware of how fortunate they are to have so much information available at the click of a mouse. Consider that researchers of the past had to physically go to libraries, search through card catalogs for possible sources of information, and wait weeks to receive books and journals that may not contain the information they needed. Then they had to begin their search all over again! Now you can find information in a matter of minutes.

Since most courses today include online components, it is assumed that you have reasonable computer skills. If you make ample use of those skills and include online research as part of your study routine, you can greatly enhance your depth of learning as well as improve your grades. Keep a web browser open as you review your course materials and laboratory assignments. When you encounter words and concepts that you have difficulty fully understanding, perform a quick web search and review as many sites as needed until the definition or concept is clear in your mind. Web searches are especially valuable in science. For example, if you have difficulty with a concept, you can usually perform an image search that will help visually clarify the object of interest. Perform a text search to find descriptions and information from leading scientists at famous institutions all over the world. For unfamiliar terms, enter the word “define” plus the unfamiliar term into your search engine and a myriad of differently phrased definitions will be available to help you. This lab manual lists numerous respected websites that you may find useful, and you will undoubtedly find many more on your own. Rely only on trusted government and educational institutions as sources for valid research data. Be especially skeptical of and double-check information garnered from personal blogs and wiki sites like wikipedia.org, where anyone, regardless of their expertise or integrity, can post and edit information. As students all over the world are finding, the worldwide web is a treasure trove of information, but not all of it is valid! Finally, while website links in this lab manual were valid at the time of printing, many good websites become unavailable or change URLs. If this happens, simply go to one of the other sites listed or perform a web search for more current sites. HOW TO PERFORM AN EXPERIMENT Although each experiment is different, the process of preparing, performing, and recording an experiment is essentially the same. Read the Entire Experiment before You Start: Knowing what you are going to do before you do it will help you organize your work and be more effective and efficient. Review Basic Safety: Before beginning work on any experiment, reread the lab manual’s safety sections, try to foresee any potential hazards, and take appropriate steps to prevent safety problems. Organize Your Work Space, Equipment, and Materials: It is hard to organize your thoughts in a disorganized environment. Assemble all required equipment and supplies before you begin working.

Outline Your Lab Notes: Outline the information needed for your Lab Notes and set up any required data tables before the experiment, to make it easier to enter observations and results as they occur. LabPaq CDs normally include a Report Assistant containing .rtf files of each experiment’s questions and data tables. These files can be copied and pasted into your Lab Notes to facilitate your compilation of data and text information. Perform the Experiment According to Instructions: Follow all directions precisely in sequential order. This is not the time to be creative. Do not attempt to improvise your own procedures! Think About What You Are Doing: Stop and give yourself time to reflect on what has happened in your experiment. What changes occurred? Why? What do they mean? How do they relate to the real world of science? This step can be the most fun and often creates "light bulb" experiences of understanding. Clean Up: Always clean your laboratory space and laboratory equipment immediately after use. Wipe down all work surfaces that may have been exposed to chemicals or dissection specimens. Blot any unused chemicals with a paper towel or flush them down the sink with generous amounts of water. Wrap dissection specimens in newspaper and plastic and place them in a sealed garbage can. Discard used pipets and other waste in your normal trash. Return cleaned equipment and supplies to their LabPaq box and store the box out of reach of children and pets. Complete Your Work: Complete your Lab Notes, answer the required questions, and prepare your Lab Report. If you have properly followed all the above steps, the conclusion will be easy.

Why Experimental Measurements Are Important:

We measure things to know something about them, to describe objects, and to understand phenomena. Experimental measurement is the cornerstone of the scientific method; thus, no theory or model of nature is tenable unless the results it predicts are measurable and in accordance with the experiment.

Your primary tasks in a science laboratory course are to create experimentally measured values, compare your results to accepted theoretical or measured values, and gain a full understanding of scientific concepts. This is true for experiments done both inside and outside of a formal laboratory. Each experiment is predicated upon a theory of scientific principle and represents a test of that theory through experimentation, observation, measurements, and analysis.

Laboratory Equipment and Techniques While many of these techniques and equipment are most applicable to specific science disciplines in formal laboratory facilities, knowledge of these items is often required for the study of other science disciplines and when working in a home laboratory. Dispensing Chemicals: To avoid contamination when pouring liquid chemicals from a reagent (ree-ey-juhnt) bottle with a glass stopper, hold the stopper in your fingers while carefully pouring the liquid into the desired container. When pouring from a screw-cap bottle, set the cap down on its top so that it does not become contaminated or contaminate anything. Be certain to put the correct cap on the bottle after use. Never pour excess chemicals back into a reagent bottle, because this may contaminate the reagents. If any liquid spills or drips from the bottle, clean it up immediately. To obtain samples of a powdered or crystalline solid from a container, it is best to pour the approximate amount of solid into a clean, dry beaker or onto a small piece of clean, creased paper for easy transport. Pour powders and crystals by tilting the container, gently shaking and rotating the solids up to the container lip, and allowing the solids to slowly fall out. If you pour too much solid, do not put any solid back in the container. Also, never put wooden splints, spatulas, or paper into a container of solids to avoid contamination. Dropping Chemicals: In micro-scale science, you use only small drops of chemicals, and it is extremely important that the drops are uniform in size and carefully observed. To ensure uniformity of drop size, use scissors to cut off the tip of the pipet perpendicular to the pipet body; cutting at an angle will distort drop sizes. Turn the pipet upside down so the dispensing chamber behind the dropper is full of liquid. Then hold the dropper in front of your eyes so you can carefully observe and count the number of drops dispensed as you slowly squeeze the pipet.

You can see the incorrect (left) and correct (right) way to dispense drops. The pipet should be held in a vertical position at eye level to ensure drops are uniform in size and the correct drops are dispensed.

Heating Chemicals: Heat solid and liquid chemicals with great care to prevent explosions and accidents.

Liquids in Beakers: To heat liquids in beakers or flasks, ensure that these containers are well supported above the heat source. Generally, the beaker or flask is placed on wire gauze supported by an iron support attached to a stand. The heat source is placed under the beaker or flask.

Liquids in Test Tubes: When heating liquids in test tubes, always use a test tube holder. Evenly heat the test tube contents by carefully moving the test tube back and forth in the flame. Heat the test tube near the top of the liquid first; heating the test tube from the bottom may cause the liquid to boil and eject from the tube.

Heating Sources for Small-scale Techniques: For micro- and small-scale science experimentation, the most commonly used heat sources are alcohol burners, candles, and burner fuel. Alcohol burners can be a problem because their flame is almost invisible, and they cannot be refilled while hot. Candles, while effective for heating small quantities of materials, tend to leave a sooty, carbon residue on the heated container that obstructs observations. Sterno and similar alcohol based fuels are very volatile and cannot be safely shipped; however, the Glycol-based fuel used in LabPaqs is safe to ship. Chafing dish (i.e., burner fuel) is actually the best of these alternatives because it has a visible flame, is easily extinguished, and does not leave excessive flame residue. Regardless of the type of burner used, never leave an ignited heat source unattended. Mass Measurement Equipment: Note that weighing scales are often called balances since weights are calculated using balance beams. Triple and quadruple beam balances are the most common measuring equipment found in laboratories. However, with today's precision technology, digital top-loading balances are becoming increasingly popular.

Triple and Quadruple Beam Scale: These balances typically include a hanging pan and vary in their degree of accuracy. After the scale has been set at zero, the object to be weighed is placed in the hanging pan, and balancing weights are added or subtracted by moving a pointer across a horizontal bar scale. When exact scale is achieved, the pointer indicates the object’s mass. Digital Top Loading Balance: This scale is initially zeroed by pressing the zero button. If your are using weighing paper or a small beaker, first tare the paper or beaker by placing it on the scale and pressing the tare button. This will produce a zero reading, and the weight of the paper or beaker will be excluded from the weighing process. Hanging Spring Scales: Measurements are taken by suspending the item from a scale, often within a container. Spring scales are not easily tared, so the container weight should be separately calculated and subtracted from the combined weight of the item and the container.

The Non-digital Analytical Balance: This instrument is very delicate, and the instructions for its use are quite detailed. Because of its extreme sensitivity, weighing on the analytical scale must be carried out in a closed chamber that is free from drafts. This instrument is seldom used by first-year science students.

Volume Measurement Equipment: To obtain accurate measurements from any glass volume measurement container, such as a beaker or graduated cylinder, you must identify and correctly read a curved surface known as the meniscus. The meniscus of water and water- based solutions concaves downward and is read at the very bottom of its curve. A mercury meniscus is convex and is read at the very top of its curve. There is no meniscus issue associated with plastic containers.

Filtration Equipment: Gravity filtration is used to remove solid precipitates or suspended solids from a mixture. It works like a small funnel or spaghetti strainer, except that it is lined with fine, conical filter paper to trap the solids. After pouring a mixture into the filter from a beaker, use a special spatula, called a rubber policeman, to scrape any remaining solids from the beaker wall into the conical filter paper. Then, use a wash bottle to rinse residue from both the beaker and rubber policeman into the filter cone to ensure that all the mixture's particles pass through the filter.

Suction filtration uses a vacuum to suck a mixture through a filter. It is much faster than but not always as efficient as gravity filtration. The required vacuum is usually created by the aspirator of a laboratory water faucet.

Bunsen Burner: This old, tried-and-true heat source relies on the combustion of natural or bottled gas. To achieve the best flame, you must properly adjust the burner's gas inlet valve and air vent. Open the valves only halfway before lighting the burner. The safest way to light the burner is to bring a lighted match to the flame opening from the side, not the top. When the burner is lit, close the air vent and adjust the gas inlet valve until the flame is approximately 10 cm high. The flame should be luminous and yellow. Next, open the air vent until the flame becomes two concentric cones. The outer cone will be faintly colored and the inner cone will be blue. The hottest part of the flame is at the tip of the blue cone. Graduated Cylinder: Graduated cylinders are available in a wide range of sizes. To read a volume in a graduated cylinder, hold the cylinder at eye level so the contents level and you can directly view the meniscus. Looking at a meniscus from below or above will create parallax and cause a false reading. Always read any scale to the maximum degree possible, including an estimate of the last digit. Buret: Burets are long, graduated tubes usually used in titration. They have a stopcock or valve on the bottom that allows you to dispense liquids in individual drops and accurately measure the quantity dispensed. Use caution when opening the stopcock to ensure that only one drop is dispensed at a time. Pipet: Pipets are small tube-type containers with openings at one end if made of plastic or at both ends if made of glass. They come in a range of volumes and are generally used to transfer specific amounts of liquids from one container to another.

Berel Pipet: These soft and flexible pipets are made of polyethylene plastic and are extensively used in LabPaqs. They have long, narrow tips and are used to deliver chemicals and to collect products. Berel pipets come in different sizes, and their tips can have different diameters and lengths. You can modify them to serve diverse purposes such as chemical scoops, gas generators, or reaction vessels. Volumetric Flask: Volumetric flasks are pear-shaped flasks with long necks used for the preparation of solutions whose concentrations need to be very accurate. Flasks come in a variety of sizes ranging from a few milliliters to several liters, and their volume levels are precisely marked. When the liquid level inside a volumetric flask is such that the meniscus lines up with the calibration mark on the neck, the volume of the liquid is exactly as stated. Unlike volumetric flasks, the markings on beakers, Erlenmeyer flasks, and most other laboratory containers are very good approximates but are not intended to be exact and precise volume measurements. Wash Bottles: These plastic squeeze bottles produce a small stream of water that can be easily dispensed as needed (e.g., washing out residue from a container). The bottles usually contain distilled or deionized water and are typically used to top off the last few milliliters of a vessel and avoid overfilling. In micro- and small-scale experimentation, plastic pipets are used for similar functions. Tissue Culture Well Plates: These microplates are plastic trays containing numerous shallow wells arranged in lettered rows and numbered columns. Similar to test tubes and beakers, you can use the wells to observe reactions, to temporarily store chemicals during experiments, and to hold pipets. The most commonly used plates are 24-well and 96-well. Distilled Water and Deionized Water: Tap water frequently contains ions that may interfere with the substances you are studying. To avoid such interference, use distilled or deionized water any time water is needed for dilution of concentration or the preparation of experimental solutions. Wash used glassware with soap, rinse with tap water, and rinse again with distilled water.

Use, Disposal, and Cleaning Instructions for Common Materials These procedures are not repeated for each experiment, because it is assumed students will always refer to them before beginning any experiment. Properly cleaning the laboratory after experimentation is a safety measure! Instrument Use

 Small quantities of chemicals are usually packaged in thin stem pipets. The drop size dispensed from small dropper bottles is different from that of the pipets. Most experiments require pipet-sized drops. It may be necessary to squeeze a few drops of chemical from a dropper bottle into a well plate, and then use a clean, empty pipet to suck up and drop the chemical.

 Once dispensed, do not return chemicals to their dropper bottles as this could cause

contamination. To avoid over-dispensing, squeeze out only a few drops of chemicals into a well plate at a time. Squeeze out more as needed.

 To use burner fuel, unscrew the cap, light the wick, and place the can under a burner

stand. Extinguish the fuel by gently placing the cap over the flame to deprive it of oxygen. Leave the cap sitting loosely on top of the wick when you are not using the fuel in order to avoid unnecessary evaporation and ensure an ample supply of fuel for all experiments. Allow the fuel to cool completely before tightly screwing on the cap for storage. If you screw the cap on while the fuel is still hot, you may create a vacuum that will make it very difficult to reopen the fuel can in the future.

 To reseal a pipet, heat the tip of a metal knife and press the pipet tip onto the hot

metal while twirling the bulb. Never simply hold a flame to the tip of the stem!

 To minimize contamination, avoid touching the surfaces of clean items that might later come in contact with test chemicals.

Storage and Disposal

 Items in LabPaq auxiliary bags are generally used multiple times or for several different experiments. Always clean and return unused auxiliary items to the bag after completing an experiment.

 Blot up used and leftover chemicals with paper towels and place in a garbage bin or

flush down a drain using copious amounts of water. The quantities of chemicals used in LabPaqs are very small and should not negatively impact the environment or adversely affect private septic systems or public sewer systems.

 Discard non-chemical experimental items with household garbage but first wrap

them in newspaper. Place these items in a securely covered trash container that cannot be accessed by children and animals.

 LabPaqs containing dissection specimens will usually contain specific information

regarding their handling. After completion of any dissecting work, wrap dissection specimens in news or waste paper, seal in a plastic bag, and place in a closed trash bin for normal garbage disposal.

Cleaning Instructions

 To clean a thin-stemmed plastic pipet, squeeze the bulb to draw up and then expel tap water from the bulb several times. Repeat this process with distilled water. Dry the pipet by repeatedly squeezing the bulb while tapping the tip on a clean paper towel. Then use gravity to help dry the pipet by forcefully swinging the pipet into a downward arch while squeezing the bulb. Lay the pipet on a clean paper towel or place it in a test tube stand and allow it to air dry.

 Use a mild liquid dishwashing detergent mixed with warm water to loosen solids or

oils that adhere to experimental glassware, plastics, and equipment and to clean laboratory equipment and the laboratory area after an experiment. Use tap water to rinse washed items well and remove all traces of detergent.

 Use a soft cloth or a test tube brush to loosen and clean residue from the surfaces of

experimental glassware, plastics, and equipment.

 Use a final rinse of distilled water to clean tap water mineral residue from newly washed items, especially beakers, cylinders, test tubes, and pipets.

 Dry test tubes by placing them upside down in the test tube rack. Air dry other items

by placing them on paper towels, aluminum foil, or a clean dishtowel. Important Notice Regarding Chemical Disposal: Due to the minute quantities and diluted and/or neutralized chemicals used in LabPaqs, the disposal methods previously described are well within acceptable levels of disposal guidelines defined for the vast majority of local solid and wastewater regulations. Since regulations occasionally vary in some communities, you are advised to check with your local area waste authorities to confirm these disposal techniques are in compliance with local regulations and/or if you should seek assistance with disposal.

HOW TO WRITE LAB NOTES AND LAB REPORTS Generally two basic records are compiled during and from scientific experimentation. The first record is your Lab Notes which you will record as you perform your experiments. Entries in your lab notebook will be the basis for your second record, the Lab Report. The Lab Report formally summarizes the activities and findings of your experiment and is normally submitted to your instructor for grading.

Lab Notes Scientists keep track of their experimental procedures and results as they work by recording Lab Notes in a journal-type notebook. In laboratories these notebooks are often read by colleagues, such as directors and other scientists working on a project. In some cases scientific notebooks have become evidence in court cases. Consequently, Lab Notes must be intelligible to others and include sufficient information so that the work performed can be replicated and there can be no doubt about the honesty and reliability of the data and the researcher. Notebooks appropriate for data recording are bound and have numbered pages that cannot be removed. Entries include all of your observations, actions, calculations, and conclusions related to each experiment. Never write data on pieces of scratch paper to transfer later, but always enter the data directly into the notebook. When you record erroneous data, neatly draw a light, diagonal line through the error, and write a brief explanation as to why you voided the data. Also record information you learn from an error. Mistakes can often be more useful than successes, and knowledge gained from them is valuable to future experimentation. As in campus-based science laboratories, independent study students are expected to keep a complete scientific notebook of their work which may or may not be periodically reviewed by the instructor. Paperbound 5x7 notebooks of graph paper work well as lab notebooks. Since it is not practical to send notebooks back and forth between instructors and students for each experiment, independent study students usually prepare formal Lab Reports and submit them along with their regular assignments to the instructor via email or fax. Lab Notes of experimental observations can be kept in many ways. Regardless of the procedure followed, the key question for deciding what kind of notes to keep is: Do I have a clear enough record that if I pick up my lab notebook or read my Lab Report in a few months, I can still explain to myself or others exactly what I did? Lab Notes generally include these components:

Title: Match the title to the title stated in the lab manual.

Purpose: Write a brief statement about what the experiment is designed to determine or demonstrate.

Procedure: Briefly summarize what you did to perform this experiment and what equipment you used. Do not simply copy the procedure statement from the lab manual.

Data Tables: Always prepare tables before experimenting, so they will be ready to receive data as it is accumulated. Tables are an excellent way to organize your observational data, and where applicable, the Procedure section advises a table format for data recording.

Observations: Record what you observed, smelled, heard, or otherwise measured? Generally, observations are most easily recorded in table form.

Questions: Thoughtfully answer the questions asked throughout and at the end of experiments. The questions are designed to help you think critically about the experiment you just performed.

Conclusions: What did you learn from the experiment? Base your conclusions on your observations during the experiment. Write your conclusions in your best, formal English, using complete sentences, full paragraphs, and correct spelling.

Some general rules for keeping a lab notebook are:

1. Leave the first two to four pages blank so you can add a Table of Contents later. Entries in the Table of Contents should include the experiment number, name, and page number.

2. Neatly write your records without being fussy. 3. Do not provide a complete Lab Report in your lab notebook. Instead, record what you

did, how you did it, and what your results were. Your records need to be substantial enough that any knowledgeable person familiar with the subject of your experiment can read the entries, understand exactly what you did, and repeat your experiment if necessary.

4. Organize all numerical readings and measurements in appropriate data tables. Refer

to the sample Lab Report in this lab manual. 5. Always identify the units (e.g., centimeters, kilograms, or seconds) for each set of data

you record. 6. Always identify the equipment you are using so you can refer to it later if you need to

recheck your work. 7. Capture the important steps and observations of your experiments using digital

photos in which you are pictured. Photos within your Lab Report document both what you observed and that you actually performed the experiment.

8. Record more rather than less data. Even details that may seem to have little bearing on your experiment (e.g., time and temperature variances when the data were taken) may turn out to have great bearing on your future results analysis.

9. Make a note if you suspect that a particular data set may not be reliable. 10. Never erase data. If you think an entry in your notes is in error, draw a single line

through it and note the correction, but don’t erase or scratch it out completely. You may later find that the information is significant after all.

Errors: Although experimental results may be in considerable error, there is never a wrong result in an experiment. Whatever happens in nature, including the laboratory, cannot be wrong. If you made your observations and measurements carefully, your results will be correct. Errors may have nothing to do with your investigation, or they may be mixed up with so many other unexpected events that your report is not useful. Even errors and mistakes have merit and often lead to our greatest learning experiences. Errors provide important results to consider; thus, you must think carefully about the interpretation of all your results, including your errors. Experiment Completion: The cardinal rule in a laboratory is to fully carry out all phases of your experiments instead of “dry-labbing” or taking shortcuts. The Greek scientist, Archytas, summed this up very well in 380 B.C.:

Lab Reports This lab manual covers the overall format that formal Lab Reports generally follow. Remember, the Lab Report should be self-contained, so anyone, including someone without a science background or lab manual, can read it, understand what was done, and understand what was learned. Data and calculation tables have been provided for many of the experiments in this lab manual, and you are encouraged to use them. Computer spreadsheet programs such as Microsoft® Excel® and websites like nces.ed.gov/nceskids/Graphing/Classic/line.asp can also greatly facilitate the preparation of data tables and graphs. Visit www.ncsu.edu/labwriter/ for additional information on preparing Lab Reports.

Lab Reports are expected to be word processed and to look organized and professional. They should be free of grammar, syntax, and spelling errors and be a respectable presentation of your work. Avoid writing in the first person as much as possible. Lab Reports should generally contain and clearly distinguish the sections discussed in detail below. The presentation and organization skills you’ll develop by producing science Lab Reports is beneficial to all potential career fields. Lab Report Format: Title Page This is the first page of the Lab Report and consists of:

A. Experiment number and/or title B. Your name C. Names of lab partner(s) D. Date and time experiment was performed E. Location if work was performed in the field F. Course number

Section 1: Abstract, Experiment, and Observation Abstract: Even though the abstract appears at the beginning of the Lab Report, you will write it last. An abstract is a very concise description of the experiment’s objectives, results, and conclusions and should be no longer than a paragraph. Experiment and Observation: In chronological order, carefully and concisely describe what was done, what was observed, and what, if any, problems were encountered. Describe what field and laboratory techniques and equipment you used to collect and analyze the data on which the conclusions are based. Insert photos and graphic illustrations in this section; graphics should be in .jpg or .gif format to minimize electronic file size. Show all your work for any calculations performed. Title every graph and clearly label the axes. Data point connections should be “best-fit curves,” which are smooth, straight or curved lines that best represent the data, instead of dot-to-dot data point connections.

Include all data tables, photos, graphs, lists, sketches, etc. in an organized fashion. Include relevant symbols and units with data. Generally one or two sentences explaining how data was obtained is appropriate for each data table.

Note any anomalies observed or difficulties encountered in collecting data as these may affect the final results. Include information about any errors you observed and what you learned from them. Be deliberate in recording your experimental procedures in detail. Your comments may also include any preliminary ideas you have for explaining the data or trends you see emerging.

Section 2: Analysis – Calculations, Graphs, and Error Analysis Generally, the questions at the end of each experiment will act as a guide when preparing your results and conclusions. The analysis is written in paragraph form and no more than one or two pages long. As you write, consider the following:

a. What is the connection between the experimental measurements taken and the final results and conclusions? How do your results relate to the real world?

b. What were the results of observations and calculations? c. What trends were noticed? d. What is the theory or model behind the experiment? e. Do the experimental results substantiate or refute the theory? Why? Be sure to refer

specifically to the results you obtained. f. Were the results consistent with your original predictions of outcomes or were you

forced to revise your thinking?

g. Did errors (e.g., environmental changes or unplanned friction) occur? If so, how did these errors affect the experiment?

h. Did any errors occur due to the equipment used (e.g., skewed estimates due to a lack

of sufficient measurement gradients on a beaker)?

i. What recommendations might improve the procedures and results? Error Analysis: In a single paragraph, comment on the accuracy and precision of the apparatuses used, include a discussion of the experimental errors, and include an estimate of the errors in your final result. Remember, errors are not mistakes. Errors arise because the apparatus and/or the environment inevitably fail to match the ideal circumstances assumed when deriving a theory or equation. The two principal sources of error are:

Physical phenomena: Elements in the environment may be similar to the phenomena being measured and may affect the measured quantity. Examples include stray magnetic or electric fields or unaccounted for friction. Limitations of the observer, analysis, and/or instruments: Examples include parallax error when reading a meter tape, the coarse scale of a graph, and the sensitivity of the instruments.

Human errors and mistakes that are not acceptable scientific errors include: calculator misuse (e.g., pushing the wrong button, misreading the display); misuse of equipment; faulty equipment; incorrectly assembled circuits or apparatuses.

Section 3: Discussion, Results, and Conclusions Discussion: Carefully organize your discussion to include consideration of the experiment’s results, interpretation of the results, and uncertainty in the results. This section is written in paragraph form and is generally no more than one to two pages in length. Occasionally it will be more appropriate to organize various aspects of the discussion differently. While not all of the following questions will apply to every experiment, consider them when writing your Lab Report. Results:

a. What is the connection among your observations, measurements, and final results? b. What were the independent or dependent variables in the experiment? c. What were the results of your calculations? d. What trends were noticeable? e. How did the independent variables affect the dependent variables? For example, did

an increase in a given independent variable result in an increase or decrease in the associated dependent variable?

Interpretation of Results:

a. What is the theory or model behind the experiment you performed? b. Do your experimental results substantiate or agree with the theory? Why or why not?

Be sure to refer specifically to your experimental results. c. Were these results consistent with your original beliefs or were you forced to

reevaluate your prior conceptions? Uncertainty in results:

a. How much did your results deviate from expected values? b. Are the deviations due to error or uncertainty in the experimental method? Can you

think of ways to decrease the amount of uncertainty? c. Are the deviations due to idealizations inherent in the theory? What factors has the

theory neglected to consider?

d. In either case, consider whether your results display systematic or random deviations.

Lab Notes and Lab Reports undoubtedly sound complex and overwhelming at first, but don’t worry. They will make more sense to you when you begin performing the experiments and writing reports. After writing your first few Lab Reports, the reports will become second nature to you. Refer to the sample Lab Report in this manual.

Laboratory Drawings Laboratory work often requires you to illustrate findings in representational drawings. Clear, well organized drawings are an excellent way to convey observations and are often more easily understood than long textual descriptions. The adage “a picture is worth a thousand words” really is true when referring to Lab Notes. Give yourself ample drawing space and leave a white margin around the actual illustration so it is clearly visible. Also leave a broad margin along one side of your drawing to insert object labels. Use a ruler to draw straight lines for the labels and connecting lines to the corresponding objects. The image below provides an example of how laboratory drawings should look when they are included in a formal Lab Report. Students often believe they can’t draw; however, with a little practice, anyone can illustrate laboratory observations. A trick many artists use is to form a mental grid over the scene and draw within the grid. For example, quickly make a free hand drawing of the diagram below. Now, mentally divide the diagram into quarters and try drawing the diagram again. In all likelihood, the second, grid-based drawing yielded a better result.

SOURCE OF DRAWING Your Name Such as MUNG BEAN Date of Drawing TITLE OF DRAWING Such as CELL STRUCTURE

Visual Presentation of Data Like pictures, good graphs and tables can quickly and clearly communicate information visually; hence, graphs and tables are often used to represent or depict collected data. Graphs and tables should be constructed to stand alone – all the information required to understand a graph or table should be included. Tables A table presents data clearly and logically. Independent data is listed in the left column and all dependent data is listed to the right. While there will be only one independent variable, there can be more than one dependent variable. The decision to present data in a table rather than a graph is often arbitrary; however, a table may be more appropriate when the data set is too small to warrant a graph or is large, complex, and not easily illustrated. Often, data tables display raw data, and a graph provides visualization of the data. Graphs A graph is composed of two basic elements: the graph itself and the graph legend. The legend provides the descriptive information needed to fully understand the graph. In the graph at right, the legend shows that the red line represents Red Delicious apples, the brown line represents Gala apples, and the green line represents Wine Sap apples. Without the legend it would be difficult to interpret this graph. When inserting a graph, choose “Scatter” as the type of graph. Trend line or Line of best fit: To more clearly show the trend between two sets of data, ”lines of best fit” or ”trend lines” are added to data. This enables us to determine the general trend of the data or to better use the data for predictive purposes. Excel or a similar spreadsheet program can easily add a trend line to the data. Use Excel to make a scatter plot of the data and then add a trend line. In most cases the line may not

Plant Height versus Fertilizer Solution X-Axis Y-Axis Fertilizer % solution

Plant Height in cm

0 25 10 34 20 44 30 76 40 79 50 65 60 40

pass through very many of the plotted points. Instead, the idea is to get a line that has equal numbers of points on either side. Most people start by viewing the data to see which trend line fits the data the best (i.e. which kind of trend line comes closest to the points). For most (but not all) of the data a linear trend line will provide a good fit. Trendlines are most useful to predict data that is not measured. In interpolation, the trend line is used to construct new data points within the range of a discrete set of known data points. Similarly, a trend line can be used to extrapolate data that are outside of the measured data set. This is illustrated in Figures 1 through 4.

Sample data set:

Time, t (seconds)

Distance, x (cm)

0.1 3.8 0.3 6.1 0.5 7.95 0.8 11

Figure 1: Sample data

Figure 2: Scatter graph of sample data

Figure 3: Scatter graph with a trendline and the equation of the line.

As an example of interpolation, if we want to know the cm-displacement at a time of 0.6 s on the Figure 4 Interpolation of data point, we add a vertical line from 0.6 s to the trendline, and then a horizontal line to the distance. This will reads an approximately distance of 9 cm. More accurately, the slope equation of the line may be used to calculate this value: y=10.182 x + 2.885; y = 10.182*0.6+2.885 = 8.99 cm To extrapolate, we would extend the trendline beyond the collected data and repeat the above process. We could also use the slope equation of the line. For example, using the equation to extrapolate the distance at 1 sec. : y = 10.182*1.0+2.885 = 13.1 cm Graph Setup: Consider a simple plot of the Plant Height versus Plant Fertilizer Concentration as shown in one of the data tables above. This is a plot of points on a set of X and Y coordinates. The X-axis or abscissa runs horizontally; the Y-axis or ordinate runs vertically. By convention, the X-axis is used for the independent variable – a manipulated variable in an experiment whose presence determines the change in the dependent variable. The Y-axis is used for the dependent variable – the variable affected by another variable or by a certain event. In this example, the amount of fertilizer is the independent variable and goes on the X-axis. The plant height, since it may change depending on changes in fertilizer amount, goes on the Y-axis. One way to determine which data goes on the X-axis versus the Y-axis

Figure 4: Interpolation of a data point

is to think about what affects what. Does fertilizer affect plant height or does plant height affect fertilizer. Only one of these options should make sense. Plant height will not change the fertilizer, but the fertilizer will affect the plant height. The variable that causes the change is independent, and the variable that changes is dependent. If the data deals with more than one dependent variable, it would be represented with three lines and a key or legend would identify which line represents which data set. In all graphs, each axis is labeled, and the units of measurement are specified. When a graph is presented in a Lab Report, the variables, the scale, and the range of the measurements should be clear. Refer to the table below when setting up a line graph.

How to Construct a Line Graph Step Explanation 1 Identify the

variables.  Independent variable: Controlled by the experimenter.

- Goes on the X-axis – the abscissa. - Located on the left side of a data chart.

 Dependent variable: Changes with the independent variable. - Goes on the Y-axis – the ordinate. - Located on the right side of a data table

2 Determine the range.

 Subtract the lowest data value from the highest. - Calculate each variable separately.

3 Determine the scale.

 Choose a scale that best fits each variable’s range (e.g., increments of one, two, five, etc.). - Choose a scale that spreads the graph over most

of the available space. 4 Number and label

each axis.  The axes tell what the graph’s data lines represent.

- Always include units of measure (e.g., days, time, meters, etc.).

5 Plot the data points.  Plot each data value on the graph with a dot. - Add the numerical data next to the dot, if there is

room and you avoid cluttering the graph. 6 Draw the graph.  Draw a straight or curved line that best fits the data

points. - Most graphs are shown as smooth lines, not dot-

by-dot connections. 7 Title the graph.  The title should clearly tell what the graph is

depicting.  Provide a legend to identify different lines, if the

graph has more than one set of data.

Computer Graphing Using MS Excel These instructions apply to the 2003 version of Excel. If you have a newer version, perform an Internet search for current instructions. This set of general instructions will be used to plot the following data:

Time, t (seconds) Distance, x (cm) 0 0 .1 9.8 .2 30.2 .3 59.9 .4 99.2 .5 148.9

When graphing x-y data, you must first determine which variable will be the X-variable and which will be the Y-variable. If you are unsure, review the previous Visual Presentation of Data section. Create a File

1. Open a blank Excel spreadsheet. 2. Save the file under an appropriate name, such as Exercise 1-Time vs

Distance.

Create Data Table

1. Enter the X-data points in the first column (A). 2. Enter the Y-data points in the second column (B).

Note: It is often useful to enter zero as the first data value, but not always. Nonetheless, it is a good habit to start.

3. Highlight all the data values by placing the curser in the first cell to be highlighted (A1) and either:

 Clicking and holding the left mouse button while pulling the mouse and

curser down and to the right so the cells are highlighted and then releasing the button.

 Holding the <Shift> key on the keyboard and using the direction arrows to move the cursor over the desired area until all cells are highlighted.

Create Graph

4. Click the Chart Wizard icon on the toolbar or select Chart from the Insert menu.

Step 1: Chart Type

5. Select XY (Scatter) from the Standard Types tab. 6. Select your preferred Chart sub-type. Although

you can choose graphs with data points, graphs with smooth lines are preferable.

7. Click Next >.

Step 2: Chart Source Data Carefully review this information to ensure the graph has the correct values for the vertical and the horizontal axes.

8. Select the Columns option button on the Data Range tab.

The range should read =Sheet1!$A$2:$B$7 This means the data:

a. Comes from Sheet 1 of the workbook. b. Comes from cells A2 through B7. c. Has been organized by data columns

instead of data rows.

9. Under the Series tab, the values for the X- and Y-axes are as follows:

X Values: =Sheet1!$A$2:$A$7 Y Values: =Sheet1!$B$2:$B$7 This means:

1. The data comes from Sheet 1 of the workbook.

2. The X-value data comes from cells A2 through A7.

3. The Y-value data comes from cells B2 through B7.

Note: If the data is reversed, replace the incorrect column letters and numbers with the correct ones.

10. To maintain the appropriate reference, rename the series of data points from the default, Series1, by entering another name in the Name field. Data is commonly used.

11. Click Next >.

Step 3: Chart Options

12. Chart Options allows you to assign titles and labels to your graph as well as determine the appearance of gridlines and legends. Make your selections.

13. Click Next >.

Step 4: Chart Location

14. Choose the location where your graph will be created.  As new sheet: Opens a new page on which the graph will appear.  As object in: Places the graph in the current spreadsheet.

If you’re unsure, select As object in so the data and graph will appear on the same page.

15. Click Finish to complete the graph.

Using Excel® To Calculate the Slope of the Line 1. Put your cursor on the line in your graph and right-click.

An option menu will drop down; select “Add Trendline.”

2. Left-click on “Add Trendline.” The window to the right will appear with icons for the types of trend lines possible.

3. For most data you will usually want a “Linear” trendline.

However, from your math classes you should recognize that the curve in your final graph (previous page) resembles a parabola which represents a quadratic or 2nd order polynomial equation. Thus, among the trendline options you will click on the polynomial option.

4. Note that a trendline has now been added to your graph as seen in the top graph of the

double graph at right. If you accidentally clicked on linear trendline, the trendline would look like the bottom graph. You can see that the linear trendline does not fit as well polynomial one.

5. Now, put your cursor on the trendline itself and right-click; then left-click on the “Format Trendline” option that appears.

6. In the box that then appears, select the “Options” tab in the Format Trendline box. Then check:

a. "Set intercept = 0", and b. “Display equation on chart”.

7. Click OK and the new graph below appears with the

equation for the trendline shown on it in the form y=mx+ b. You should recognize that m = slope. (Caveat: Only click on “select intercept = 0” when the line goes through zero.)

SHORTCUT: You can select the type and formatting of a trendline in one step. From Step 3 after selecting the polynomial option, go straight to the Options tab where you can immediately check “Set intercept = 0 “and “Display equation on chart”. Click OK and you are done. Delete your graph and start over to practice and feel comfortable with all the above graphing steps and this shortcut.

Using Excel® for Calculations Many times in physics and scientific or engineering work, one must repeat the same basic calculations using different sets of numbers. As an example consider trying to find the average speed between the distances in our earlier table. The first calculation would look like … Vavg = ∆x = 9.8 cm – 0.0 cm = 98 cm/s ∆t 0.1 – 0.0 s

The second calculation would be Vavg = ∆x = 30.2 cm – 9.8 cm = 204 cm/s ∆t 0.2 – 0.1 s There are only five calculations to compute here and doing all five on a calculator is not a lot of work. However, if there were 100 or 1000 such calculations, it would be extremely laborious! Fortunately Excel® can do these calculations easily and quickly with formulas plus copy and paste functions. Let’s try it with the above data. First, enter the time and distance data into Excel®. Start by inputting the zero values in “cells” A1 and B1 and then enter the rest of the data in the A and B “columns.” (Hint: you may wish to begin inputting your data in “row” 5 or so in the future in order to leave space above the data to later include a spreadsheet title or other information.) Observe that the first non-zero data is in row 2 and in cells A2 and B2. In addition, our time data [t] is in column A and our distance data [x] is in column B. Next, think about how you might construct a formula for our problem. If ∆x, the change in distance can be computed as B2-B1, and ∆t, the change in time can be computed as A2-A1, then we could use this formula for the change in distance over the change in time: = (B2-B1)/(A2-A1) This formula is a math statement that says the difference of the values in boxes B3 and B2 should be divided by the difference of the values in boxes A3 and A2. In order to record in column C the average speed between the distances for all of the sets of data, you must first input the formula above into cell C2 and then copy and paste it into the remaining cells. To do this: 1. Place your curser in the cell C2 2. Insert an equal sign [=] to alert Excel® this will be a formula rather than data.

Time (seconds) Distance (cm) 0 0 .1 9.8 .2 30.2 .3 59.9 .4 99.2 .5 148.9

3. Type your = (B2-B1)/(A2-A1) formula using no extra spaces between numbers and operative signs and hit enter. Observe that your formula appears in the fx box above the columns as it is entered. Refer to this box to make sure your formula is typed correctly.

4. If your formula was correctly input Excel® will do the calculation for you and a value of 98 should appear in C2.

You can now use the same formula for each of the successive sets of values and simplify the process by using the copy and paste functions. When you copy a formula from one Excel® cell and then paste it in another cell, Excel® automatically adjusts the formula to correspond to the cursors’ new position. To copy and paste data from a cell, move your cursor to that cell and either use:

 Your mouse: right click to display options and click copy or paste  Edit command at the top of your screen: select the copy or paste option and left

click the mouse or hit enter on the keyboard.

 Keyboard commands: use Control + C for copy and Control + V for paste 5. Move your cursor to cell C2 and “copy” its contents in one of the ways described above.

When the cell appears to vibrate, its contents can be copied into other cells. Now move the cursor to 3C and “paste” it in one of the ways described above. To stop the source cell vibrations and end the possibility of copying its data further, hit enter or escape. Note that the formula for cell C3 now correctly reads =(B3-B2)/(A3-A2) and corresponds to the data in row 3.

6. To transfer the formula into multiple cells at the same time, copy the formula in cell C2, highlight cells C3 through C6, and paste. Note that the formula adjusts itself for each row of data and Excel® will properly calculate the average velocity for each of the additional sets of data; the answers are now shown in column C.

Reinforcing Exercise: For the set of data values located on the next page, find the average acceleration between each of the speeds using Excel®.

Time (sec) Speed (cm/s) 0.0906 138 0.1361 184 0.1714 219 0.201 249 0.2267 275 0.2281 274 0.2511 299 0.262 310 0.2916 340 0.3075 355 0.3187 364 0.3371 386 0.3417 390 0.3642 408 0.3723 422 0.3872 435 0.3994 441 0.4224 471 0.452 499

SAFETY CONCERNS You, as a responsible science student and researcher, are solely responsible for safely storing and using your LabPaq materials and for conducting your experiments in a safe and responsible manner. Items in your LabPaq can be especially dangerous to children and pets, so the LabPaq should always be kept safely stored out of their reach. The LabPaq may contain acids or other chemicals that can cause burns if mishandled plus serious illness and or death if consumed. Many LabPaq items are made of glass and/or have sharp edges that pose potential risks for cuts and scratches. While LabPaq thermometers do not contain mercury, they might still break and cause injury. LabPaqs contain small items and materials that could cause choking, injury, or death if misused. Experimentation may require you to climb, push, pull, spin, and whirl. While these activities are not necessarily dangerous, they can pose hazards, and you should always undertake these activities cautiously and with consideration for your surroundings. If you need to climb to take measurements, make sure any stool, chair, or ladder you use is sturdy and take ample precautions to prevent falls. It is wise to have a partner help keep you stable when you must climb. Be especially aware of experimental equipment that you must put in motion, and act cautiously to ensure that items cannot go astray and cause injury to people or property. If you or anyone accidentally consumes or otherwise comes into contact with a substance that could be toxic or cannot be easily washed away, immediately call:

The National Poison Control Center: 1-800-222-1222

Your eyesight is precious and should be protected against chemical spills or splashes as well as flying objects and debris. Always wear safety goggles when working with chemicals of any kind and when working with non-chemical objects that could possibly fly into your eyes. Since chemicals, dirt, and germs are often involved in laboratory experiments, you should never eat or smoke in your laboratory area. Protect your body by keeping your hair tied back from your face and by wearing old clothing that fully covers your arms and legs.

You also need to protect your home furnishings from damage during your experimentation. Cover your work surface with plastic or paper towels when appropriate to prevent ruining furniture and to aid in cleanup. The best safety tools you have are your own mind and intellectual ability to think and plan. After previewing each experiment, carefully think about what safety precautions you need to take to experiment safely, and then take them! Since it is impossible to control students’ use of this lab manual and related LabPaqs or students’ work environments, the author(s) of this lab manual, the instructors and institutions that adopt it, and Hands-On Labs, Inc. – the publisher of the lab manual and the producer of LabPaqs – authorize the use of these educational products only on the express condition that the purchasers and users accept full and complete responsibility for all and any liability related to their use of same. Additional terms authorizing the use of a LabPaq are contained in its Purchase Agreement available at www.LabPaq.com.

Basic Safety Guidelines This section contains vital information that you must thoroughly read and completely understand before beginning to perform experiments. Science experimentation is fun but involves potential hazards which you must acknowledge to avoid. To safely conduct science experiments, you must learn and follow basic safety procedures. While there may be fewer safety hazards for physics and geology experimentation than chemistry and biology, safety risks exist in all science experimentation and should be taken very seriously. Thus, the following safety procedures review is relevant to all students regardless of their field of study While this lab manual tries to include all relevant safety issues, not every potential danger can be foreseen, as each experiment involves different safety considerations. You must always act responsibly, learn to recognize potential dangers, and always take appropriate precautions. Regardless of whether you will be working in a campus or home laboratory setting, it is extremely important that you know how to anticipate and avoid possible hazards and to be safety conscious at all times.

Basic Safety Procedures Science experimentation often involves using toxic chemicals, flammable substances, breakable items, and other potentially dangerous materials and equipment. All of these things can cause injury and even death if not properly handled. These basic safety procedures apply when working in a campus or home laboratory.  Because eyesight is precious and eyes are vulnerable to chemical spills and splashes,

shattered rocks and glass, and floating and flying objects: - Always wear eye protecting safety goggles when experimenting.

 Because toxic chemicals and foreign matter may enter the body through digestion:

- Never drink or eat in laboratory areas. - Always wash your hands before leaving the laboratory. - Always clean the laboratory area after experimentation.

 Because toxic substances may enter the body through the skin and lungs:

- Ensure the laboratory always has adequate ventilation. - Never directly inhale chemicals. - Wear long-sleeved shirts, pants, and enclosed shoes when in the laboratory. - Wear gloves and aprons when appropriate.

 Because hair, clothing, and jewelry can create hazards, cause spills, and catch fire while

experimenting: - Always tie or pin back long hair. - Always wear snug fitting and preferably old clothing. - Never wear dangling jewelry or objects.

 Because a laboratory area contains various fire hazards:

- Smoking is always forbidden in laboratory areas.  Because chemical experimentation involves numerous potential hazards:

- Know how to locate and use basic safety equipment. - Never leave a burning flame or reaction unattended. - Specifically follow all safety instructions. - Never perform any unauthorized experiments. - Always properly store equipment and supplies.

 Because science equipment and supplies often include breakable glass and sharp items

posing potential risks for cuts and scratches; and small items and dangerous chemicals potentially causing death or injury if consumed:

- Carefully handle all science equipment and supplies. - Keep science equipment and supplies stored out of the reach of pets and small

children. - Ensure pets and small children will not enter the lab area while experimenting.

 Because science experimentation may require students to climb, push, pull, spin, and whirl:

- Undertake these activities cautiously and with consideration for the people, property, and objects that could be impacted.

- Ensure stools, chairs, or ladders used to climb are sturdy and take ample precautions to prevent falls.

 Because your best safety tools are your own mind and intellectual ability:

- Always preview each experiment, carefully think about what safety precautions need to be taken to experiment safely, and then take them.

Basic Safety Equipment: You can find the following pieces of basic safety equipment in all campus laboratories. Informal and home laboratories may not have all of these items, but you can usually make simple substitutions. You should know the exact location and proper use of these items.

Eyewash Station: All laboratories should have safety equipment to wash chemicals from the eyes. A formal eyewash station looks like a water fountain with two faucets directed up at spaces to match the space between the eyes. In case of an accident, the victim's head is placed between the faucets while the eyelids are held open, so the faucets can flush water into the eye sockets and wash away the chemicals. In an informal laboratory, you can substitute a hand-held shower wand for an eyewash station. After the eyes are thoroughly washed, consult a physician promptly.

Fire Blanket: A fire blanket is a tightly woven fabric used to smother and extinguish a fire. It can cover a fire area or be wrapped around a victim who has caught on fire.

Fire Extinguisher: There are several types of fire extinguishers, but at least one should be available in all laboratories. You should familiarize yourself with and know how to use the particular fire extinguisher in your laboratory. At a minimum, home laboratories should have a bucket of water and a large container of sand or dirt to smother fires.

First-Aid Kit: This kit of basic first-aid supplies is used for the emergency treatment of injuries and should be standard in both formal and informal laboratories. It should always be well stocked and easily accessible.

Fume Hood: A fume hood is a hooded area containing an exhaust fan that expels noxious fumes from the laboratory. Experiments that might produce dangerous or unpleasant vapors are conducted under this hood. In an informal laboratory such experiments should be conducted only with ample ventilation and near open windows or doors. If a kitchen is used for a home laboratory, the exhaust fan above the stove substitutes nicely for a fume hood.

Safety Shower: This shower is used in formal laboratories to put out fires or douse people who have caught on fire or suffered a large chemical spill. A hand-held shower wand is the best substitute for a safety shower in a home laboratory.

Safety Goggles: There is no substitute for this important piece of safety equipment! Spills and splashes do occur, and eyes can very easily be damaged if they come in contact with laboratory chemicals, shattered glass, swinging objects, or flying rock chips. While normal eyeglasses provide some protection, objects can still enter the eyes from the side. Safety goggles cup around all sides of the eyes to provide the most protection and can be worn over normal eyeglasses when necessary.

Spill Containment Kit: This kit consists of absorbent material that can be ringed around a spilled chemical to keep the spill contained until it can be neutralized. The kit may simply be a container full of sand or other absorbent material such as cat litter.

Potential Laboratory Hazards: Recognizing and respecting potential hazards is the first step toward preventing accidents. Please appreciate the grave dangers the following laboratory hazards represent. Work to avoid these dangers and consider how to respond properly in the event of an accident. Acid Splatter: When water is added to concentrated acid, the solution becomes very hot and may splatter acid. Splattering is less likely to occur if you add acid slowly to the water. Remember this AAA rule: Always Add Acid to water, never add water to acid. Chemical Ingestion: Virtually all chemicals found in a laboratory are potentially toxic. To avoid ingesting dangerous chemicals, never taste, eat, or drink anything while in the laboratory. All laboratories, and especially those in home kitchens, should always be thoroughly cleaned after experimentation to avoid this hazard. In the event of any chemical ingestion, immediately consult a physician. Chemical Spills: Flesh burns may result if acids, bases, or other caustic chemicals are spilled and come in contact with skin. Flush the exposed skin with a gentle flow of water for several minutes at a sink or safety shower. Neutralize acid spills with sodium bicarbonate – simple baking soda. If eye contact is involved, use the eyewash station or its substitute. Use the spill containment kit until the spill is neutralized. To better protect the body from chemical spills, wear long-sleeved shirts, full-length pants, and enclosed shoes when in the laboratory. Fires: The open flame of a Bunsen burner or any heating source, combined with inattention, may result in a loose sleeve, loose hair, or some unnoticed item catching fire. Except for water, most solvents, including toluene, alcohols, acetones, ethers, and acetates, are highly flammable and should never be used near an open flame. As a general rule, never leave an open flame or reaction unattended. In case of fire, use a fire extinguisher, fire blanket, and/or safety shower. Fume Inhalation: To avoid inhaling dangerous fumes, partially fill your lungs with air and, while standing slightly back from the fumes, use your hand to waft the odors gently toward your nose. Lightly sniff the fumes in a controlled fashion. Never inhale fumes directly! Treat inhalation problems with fresh air, and consult a physician if the problem appears serious. Glass Tubing Hazards: Never force a piece of glass tubing into a stopper hole. The glass may snap, and the jagged edges can cause serious cuts. Before inserting glass tubing into a rubber or cork stopper hole, be sure the hole is the proper size. Lubricate the end of the glass tubing with glycerol or soap, and then, while grasping the tubing with a heavy glove or towel, gently but firmly twist it into the hole. Treat any cuts with appropriate first aid.

Heated Test Tube Splatter: Splattering and eruptions can occur when solutions are heated in a test tube. You should never point a heated test tube towards anyone. To minimize this danger, direct the flame toward the top rather than the bottom of the test tube. Gently agitate the tube over the flame to heat the contents evenly.

Horseplay: A laboratory full of potentially dangerous chemicals and equipment is a place for serious work, not for horseplay! Fooling around in the laboratory is an invitation for an accident. Shattered Glassware: Graduated cylinders, volumetric flasks, and certain other pieces of glassware are not designed to be heated. If heated, glassware is likely to shatter and cause injuries. Always ensure you are using heatproof glass before applying it to a heat source. Take special caution when working with any type of laboratory glassware CAUTION for Women: If you are pregnant or could be pregnant, you should seek advice from your personal physician before doing any type of science experimentation.

Material Safety Data Sheets An important skill in the safe use of chemicals is the ability to read a Material Safety Data Sheet (MSDS). An MSDS is designed to provide chemical, physical, health, and safety information on chemical reagents and supplies. It provides information about how to handle, store, transport, use and dispose of chemicals in a safe manner. An MSDS also provides workers and emergency personnel with the proper procedures for handling and working with chemical substances. While there is no standard format for an MSDS, any MSDS provides basic information about physical data, toxicity, health effects, first-aid procedures, chemical reactivity, safe storage, safe disposal, required protective equipment, and spill cleanup procedures. An MSDS is required to be readily available at any business where any type of chemical is used. Even daycare centers and grocery stores need MSDSs for their cleaning supplies. It is important to know how to read and understand an MSDS. An MSDS is generally organized into the following sections:

Section 1: Product Identification Chemical name and trade names Section 2: Hazardous Ingredients Components and percentages Section 3: Physical Data Boiling point, density, solubility in water, appearance, color, etc. Section 4: Fire and Explosion Data Flash point, extinguisher media, special fire fighting procedures, and unusual fire and explosion hazards Section 5: Health Hazard Data Exposure limits, effects of overexposure, emergency and first-aid procedures

Section 6: Reactivity Data Stability, conditions to avoid, incompatible materials, etc. Section 7: Spill or Leak Procedures Steps to take to control and clean up spills and leaks and waste disposal methods Section 8: Control Measures Respiratory protection, ventilation, protection for eyes or skin, or other needed protective equipment

Section 9: Special Precautions How to handle and store, steps to take in a spill, disposal methods, and other precautions

The MSDS is a tool available to employers and workers for making decisions about chemicals. The least hazardous chemical should be selected for use whenever possible, and procedures for storing, using, and disposing of chemicals should be written and communicated to workers. View MSDS information at www.hazard.com/msds/index.php. You can also find a link to MSDS information at www.LabPaq.com. If there is ever a problem or question about the proper handling of any chemical, seek information from one of these sources.

Safety Quiz Refer to the illustration on the following page when answering the questions.

1. List three (3) unsafe activities in the illustration and explain why each is unsafe.

2. List three (3) correct procedures depicted in the illustration.

3. What should Tarik do after the accident?

4. What should Lindsey have done to avoid an accident?

5. Compare Ming and David's laboratory techniques. Who is following the rules?

6. What are three (3) things shown in the laboratory that should not be there?

7. Compare Joe and Tyler's laboratory techniques. Who is working the correct way?

8. What will happen to Ray and Chris when the instructor catches them?

9. List three (3) items in the illustration that are there for the safety of the students.

10. What is Consuela doing wrong?

Science Lab Safety Reinforcement Agreement Any type of science experimentation involves potential hazards, and unforeseen risks may exist. The need to prevent injuries and accidents cannot be overemphasized! Use of this lab manual and any LabPaqs are expressly conditioned upon your agreement to follow all safety precautions and accept full responsibility for your actions. Study the safety section of this lab manual until you can honestly state the following:  Before beginning an experiment I will first read all directions and then assemble and

organize all required equipment and supplies.  I will select a work area that is inaccessible to children and pets while experiments are

in progress. I will not leave experiments unattended, and I will not leave my work area while a chemical equipment is set up unless the room is locked.

 To avoid the potential for accidents, I will clear my home laboratory workspace of all

non-laboratory items before setting up equipment and supplies for my experiments.  I will never attempt an experiment until I fully understand it. If in doubt about any part

of an experiment, I will first speak with my instructor before proceeding.  I will wear safety goggles when working with chemicals or items that can get in my

eyes  I know that except for water, most solvents, such as toluene, alcohols, acetone,

ethers, and ethyl acetate are highly flammable and should never be used near an open flame.

 I know that the heat created when water is added to concentrated acids is sufficient

to cause spattering. When preparing dilute acid solutions, I will always add the acid to the water – rather than the water to the acid – while slowly stirring the mixture.

 I know it is wise to wear rubber gloves and goggles when handling acids and other

dangerous chemicals; I should neutralize acid spills with sodium bicarbonate; and I should wash acid spilled on skin or clothes immediately with plenty of cold water.

 I know that many chemicals produce toxic fumes, and cautious procedures should be

used when smelling any chemical. When I wish to smell a chemical, I will never hold it directly under my nose, but will use my hand to waft vapors toward my nose.

 I will always handle glassware with respect and promptly replace any defective

glassware. Even a small crack can cause glass to break, especially when heated. To avoid cuts and injuries, I will immediately dispose of any broken glassware.

 I will avoid burns by testing glass and metal objects for heat before handling. I know that the preferred first aid for burns is to immediately hold the burned area under cold water for several minutes.

 I know that serious accidents can occur when wrong chemicals are used in an

experiment. I will always read labels before removing chemicals from their containers.  I will avoid the possibility of contamination and accidents by never returning an

unused chemical to its original container. To avoid waste I will try to pour only the approximate amount of chemicals required.

 I know to immediately flush any chemical spill on the skin with cold water and consult

a doctor if required.  To protect myself from potential hazards, I will wear long pants, a long-sleeved shirt,

and enclosed shoes when performing experiments. I will tie up any loose hair, clothing, or other materials as well.

 I will never eat, drink, or smoke while performing experiments.

 After completing all experiments I will clean my work area, wash my hands, and store

the laboratory equipment in a safe place inaccessible to children and pets.  I will always conscientiously work in a reasonable and prudent manner to optimize my

safety and the safety of others whenever and wherever I am involved with any type of science equipment or experimentation.

It is impossible to control students’ use of this lab manual and related LabPaqs or students’ work environments. The author(s) of this lab manual, the instructors and institutions that adopt it, and Hands-On Labs, Inc. – the publisher of the lab manual and producer of LabPaqs – authorize the use of these educational products only on the express condition that the purchasers and users accept full and complete responsibility for all and any liability related to their use of same. Please review this document several times until you are certain you understand it and will fully abide by its terms. Then sign and date the agreement were indicated. I am a responsible adult who has read, understands, and agrees to fully abide by all safety precautions prescribed in this lab manual for laboratory work and for the use of a LabPaq. Accordingly, I recognize the inherent hazards associated with science experimentation; I will always experiment in a safe and prudent manner; and I unconditionally accept full and complete responsibility for any and all liability related to my purchase and/or use of a science LabPaq or any other science products or materials provided by Hands-On Labs, Inc. (HOL). ____________________________________________________ ____________ Student’s Name (print) and Signature Date

EXPERIMENTS

Experimental Errors and Uncertainty

Experiment Summary:

Students will have the opportunity to learn about accuracy, precision, and types and sources of experimental error. They will learn the importance of

significant figures as well as how to calculate percent error, percent difference, and standard and mean deviation. Students will use a free fall

experiment to learn how to report measurement results, graph data, determine the slope of a line, and calculate gravitational force and percent

error.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To gain an understanding of experimental errors and uncertainty.

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 1 Pen and pencils 1 Paper, plain and graph 1 Computer and spreadsheet program

Discussion and Review

No physical quantity can be measured with perfect certainty; there are always errors in any measurement. This means that if we measure some quantity and then repeat the measurement we will almost certainly measure a different value the second time. How then can we know the “true” value of a physical quantity? The short answer is that we cannot. However, as we take greater care in our measurements and apply ever more refined experimental methods we can reduce the errors and thereby gain greater confidence that our measurements approximate ever more closely the true value. “Error analysis” is the study of uncertainties in physical measurements. A complete description of error analysis would require much more time and space than we have in this course. However, by taking the time to learn some basic principles of error analysis we can:

 Understand how to measure experimental error;  Understand the types and sources of experimental errors;  Clearly and correctly report measurements and the uncertainties in measurements;

and

 Design experimental methods and techniques plus improve our measurement skills to reduce experimental errors.

Two excellent references on error analysis are:

 John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2d Edition, University Science Books, 1997; and

 Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992.

Accuracy and Precision Experimental error is the difference between a measurement and the true value or between two measured values. Experimental error itself is measured by its accuracy and precision. Accuracy measures how close a measured value is to the true value or accepted value. Since a true or accepted value for a physical quantity may be unknown, it is sometimes not possible to determine the accuracy of a measurement. Precision measures how closely two or more measurements agree with each other. Precision is sometimes referred to as “repeatability” or “reproducibility”. A measurement that is highly reproducible tends to give values which are very close to each other.

Figure 1 defines accuracy and precision with an analogy of the grouping of arrows in a target. Figure 1: Accuracy vs. Precision

Types and Sources of Experimental Errors When scientists refer to experimental errors they are not referring to what are commonly called mistakes, blunders, or miscalculations or sometimes illegitimate, human or personal errors. Personal errors can result from measuring a width when the length should have been measured, or measuring the voltage across the wrong portion of an electrical circuit, or misreading the scale on an instrument, or forgetting to divide the diameter by 2 before calculating the area of a circle with the formula A = π r2. Such errors are certainly significant but they can be eliminated by performing the experiment again correctly the next time. On the other hand, experimental errors are inherent in the measurement process. They cannot be eliminated simply by repeating the experiment, no matter how carefully. There are two types of experimental errors: systematic errors and random errors. Systematic Errors: Systematic errors are errors that affect the accuracy of a measurement. Systematic errors are “one-sided” errors because, in the absence of other types of errors, repeated measurements yield results that differ from the true or accepted value by the same amount. The accuracy of measurements subject to systematic errors cannot be improved by repeating those measurements. Systematic errors cannot easily be analyzed by statistical analysis. Systematic errors can be difficult to detect, and once detected they can only be reduced by refining the measurement method or technique. Common sources of systematic errors are faulty calibration of measuring instruments, poorly maintained instruments, or faulty reading of instruments by the user. A common form of this last source of systematic error is called “parallax error”, which results from the user reading an instrument at an angle resulting in a reading which is consistently high or consistently low.

Random Errors: Random errors are errors that affect the precision of a measurement. Random errors are “two-sided” errors because, in the absence of other types of errors, repeated measurements yield results that fluctuate above and below the true or accepted value. Measurements subject to random errors differ from each other due to random, unpredictable variations in the measurement process. The precision of measurements subject to random errors can be improved by repeating those measurements. Random errors are easily analyzed by statistical analysis. Random errors can be detected and reduced by repeating the measurement or by refining the measurement method or technique. Common sources of random errors are problems estimating a quantity that lies between the graduations (the measurement lines) on an instrument and the inability to read an instrument because the reading fluctuates during the measurement. Calculating Experimental Error When a scientist reports the results of an experiment the report must describe the accuracy and precision of the experimental measurements. Some common ways to describe accuracy and precision are described below. Significant Figures: The least significant digit in a measurement depends on the smallest unit that can be measured using the measuring instrument. The precision of a measurement can then be estimated by the number of significant digits with which the measurement is reported. In general, any measurement is reported to a precision equal to 1/10 of the smallest graduation on the measuring instrument, and the precision of the measurement is said to be 1/10 of the smallest graduation. For example, a measurement of length using a meter tape with 1-mm graduations will be reported with a precision of ±0.1 mm. A measurement of volume using a graduated cylinder with 1 mL graduations will be reported with a precision of ±0.1 mL. Digital instruments are treated differently. Unless the instrument manufacturer indicates otherwise, the precision of measurement made with digital instruments are reported with a precision of ±½ of the smallest unit of the instrument. For example, a digital voltmeter reads 1.493 volts; the precision of the voltage measurement is ±½ of 0.001 volts or ±0.0005 volt. Percent Error: Percent error measures the accuracy of a measurement by the difference between a measured or experimental value E and a true or accepted value A. The percent error is calculated from the following equation:

Equation 1 % Error = | E – A| x 100% A

Percent Difference: Percent difference measures precision of two measurements by the difference between the measured or experimental values E1 and E2 expressed as a fraction of the average of the two values. The equation used to calculate the percent difference is: Equation 2 Mean and Standard Deviation: When a measurement is repeated several times we see the measured values are grouped around some central value. This grouping or distribution can be described with two numbers: the mean, which measures the central value and the standard deviation, which describes the spread or deviation of the measured values about the mean. For a set of N measured values for some quantity x, the mean of x is represented by the symbol <x> and is calculated by the following formula: Equation 3 Where xi is the i-th measured value of x. The mean is simply the sum of the measured values divided by the number of measured values. The standard deviation of the measured values is represented by the symbol σx and is given by the formula: Equation 4 The standard deviation is sometimes referred to as the “mean square deviation.” It measures how widely spread the measured values are on either side of the mean. The meaning of the standard deviation can be seen from the figure on the right. This is a plot of data with a mean of 0.5. As shown in this graph, the larger the standard deviation, the more widely spread the data is about the mean. For measurements that have only random errors the standard deviation shows that 68% of the measured values are within σx from the mean, 95% are within 2σx from the mean, and 99% are within 3σx from the mean.

Reporting the Results of an Experimental Measurement When a scientist reports the result of an experimental measurement of a quantity x, that result is reported with two parts. First, the best estimate of the measurement is reported. The best estimate of a set of measurement is usually reported as the mean <x> of the measurements. Second, the variation of the measurements is reported. The variation in the measurements is usually reported by the standard deviation σx of the measurements. The measured quantity is then known to have a best estimate equal to the average, but it may also vary from <x>+ σx to <x> - σx. Any experimental measurement should then be reported in the following form: x = <x>  x Example: Consider Table 1 below that lists 30 measurements of the mass m of a sample of some unknown material. Table 1: Measured Mass (kg) of Unknown We can represent this data on a type of bar chart called a histogram (Figure 2), which shows the number of measured values which lie in a range of mass values with the given midpoint.

Figure 2: Mass of Unknown Sample For the 30 mass measurements the mean mass is given by: <m> = 1/30 (33.04 kg) = 1.10 kg We see from the histogram that the data does appear to be centered on a mass value of 1.10 kg. The standard deviation is given by: We also see from the histogram that the data does, indeed, appear to be spread about the mean of 1.10 kg so that approximately 70% (= 20/30x100) of the values are within σm from the mean. The measured mass of the unknown sample is then reported as:

m = 1.10± 0.05 kg PROCEDURE: The data table that follows shows data taken in a free-fall experiment. Measurements were made of the distance of fall (Y) at each of the four precisely measured times. From this data perform the following: 1. Complete the table. 2. Plot a graph <y> versus t (plot t on the abscissa, i.e., x-axis).

3. Plot a graph <y> versus t2 (plot t2 on the abscissa, i.e., x-axis). The equation of motion for

an object in free fall starting from rest is y = ½ gt2, where g is the acceleration due to gravity. This is the equation of a parabola, which has the general form y = ax2.

4. Determine the slope of the line and compute an experimental value of g from the slope

value. Remember, the slope of this graph represents ½ g. 5. Compute the percent error of the experimental value of g determined from the graph in

step 4. (Accepted value of g = 9.8 m/s2) 6. Use a spreadsheet to perform the calculations and plot the graphs indicated.

Time, t (s)

Dist. y1 (m)

Dist. y2 (m)

Dist. y3 (m)

Dist. y4 (m)

Dist. y5 (m)

<y> σ t2

0 0 0 0 0 0 0.5 1.0 1.4 1.1 1.4 1.5 0.75 2.6 3.2 2.8 2.5 3.1 1.0 4.8 4.4 5.1 4.7 4.8 1.25 8.2 7.9 7.5 8.1 7.4

Measurement: Length, Mass, Volume, Density,

and Time

Experiment Summary:

Students will have the opportunity to learn how to make basic distance, mass, density, and time measurements using the metric system. They will

calculate volume and density; use spreadsheet software to graph circumference and diameter relationships; estimate and measure distance,

time, and mass; measure the circumference of various round objects, plot the data, calculate the slope, and determine the value of ; use Archimedes’

principle to determine density; and calculate percent error.

Peter Jeschofnig, Ph.D. Version 09.1.03

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before beginning. Take time to organize the materials needed and set aside

a safe workspace in which to complete the exercise.

Objectives

 To make basic distance, mass, density, and time measurements

 To make calculations of volume and density, using proper units

 To practice graphing the relationship between the circumference of a

circle and its diameter using spreadsheet software

Estimated time to perform this experiment: 3 hours.

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 3 Box-like objects (block, thick book, shoebox, etc.)

2 Pencils or pens 1 Chair or step stool 5 Circular objects of different size (cups,

plates, etc.) 1 Computer and spreadsheet program 1

1 Lab partner Cup filled with tap water

From LabPaq

1 Cylinder, 25 mL 1 Ruler, Metric 1 Scale-Spring-10-g in Box Labeled Slim Pen

Scale 1 Scale-Spring-500-g 1 Stopwatch-digital 1 Tape measure, 1.5-m 1 Tape measure, 3-m Marble, Bolt, & Spring Bag

1 Bolt, Metal - Small

String & Weight Bag 1 String - Qty-4.0 Meters

Discussion and Review

Physics is a quantitative experimental science based on measurement. In the physics laboratory, it is important to know how to measure fundamental quantities like length, mass, and time with precision and accuracy. In this experiment, you will study techniques for using several pieces of laboratory equipment and the units of measurements most frequently used in a laboratory. Scientific measurements normally use metric units:

LENGTH: The meter (m) is the basic SI (Systeme International) unit of length. A meter is just a little longer than a yard. Since 1983, the meter has been defined as the distance travelled by light in a vacuum in 1⁄299,792,458 of a second.

1 in = 2.54 cm 1000 m = 1 km 1 km = 0.621 mi 1 m = 100 cm 1 m = 1.09 yd 1 m = 1000 mm 1 m = 3.281 ft 1 cm = 10 mm

Figure 1 – Meterstick measurements.

The International System of Units or SI (from the French Système international d'unités) is a modern form of the metric system based on units of ten. This system has been adopted

throughout the world with the exception of the United States, Liberia, and Burma (Myanmar).

TEMPERATURE: Temperature measurements are usually taken in degrees Celsius (°C). SI temperatures are expressed using the Kelvin (K) scale. Use the following formulas to convert between Fahrenheit (°F), Celsius, and Kelvin scales.

K CT T  273 F CT ( . )T 1 8 32

F C

(T ) T

. 

 32

1 8

Figure 2 – Comparison of different temperature scales VOLUME: The basic unit of volume used in the science lab is the liter (L), which is slightly larger than a quart. Related to the liter is the milliliter (mL), which is one-thousandth of a liter (0.001 L). A milliliter is equal to the volume of a cube that measures 1 cm on each

side. Because the volume of a cube is equal to the length multiplied by the width multiplied by the height, the volume of a cube that measures 1 cm on each side is equal to 1 cm3.

1 L = 1000 mL = 1000 cm3

1 mL = 1 cm3 1 L = 1.06 qt

When reading a glass graduated cylinder, read the volume at the bottom of the meniscus. The meniscus is the curve in a liquid due to the attraction between the glass and the water. In Figure 3, the meniscus is marked with an arrow. The volume equals 11.5 mL in this 25- mL graduated cylinder.

Figure 3 – Graduated cylinder with meniscus.

MASS: The kilogram (kg) is the SI unit of mass and equals about 2.2 pounds (lb.). In the laboratory, we usually work with the gram (g), which represents one-thousandth of a kilogram, and with the milligram (mg), which equals one-thousandth of a gram.

1 kg = 1000 g 1 lb = 454 g 1 g = 1000 mg 1 kg = 2.20 lb

NOTE: Mass and weight are not the same thing! Mass is a quantity of matter while weight refers to the gravitational force of attraction exerted upon an object. In the laboratory, mass measurement will be used, and the verb "weigh" will only be used to instruct you to determine the mass of an object.

Figure 4 – A kilogram weight located at the Danish National Metrology Institute. SI standards are primarily located at the International Bureau of Weights and Measures’ House of Breteuil on the outskirts of Paris.

TIME: The second (s) is the basic SI unit of time. DENSITY: The density ρ of a substance is its mass (m) per unit volume (V). The densities of liquids are usually reported in grams per milliliter (g/mL). The densities of solids are usually reported in grams per cubic centimeter (g/cm3 or g/cc). The density of water is 1 g/mL; thus the mass of one liter of water is one kilogram. Substances with densities less than 1 g/mL will float on water.

Equation 1: m V

 

Example Densities: Water = 1 g/mL Aluminum = 2.70 g/cm3

Iron = 7.85 g/cm3 Lead = 11.35 g/cm3 Gold = 19.30 g/cm3 SPECIFIC GRAVITY (S.G.) or RELATIVE DENSITY of a substance is the ratio of its density to the density of water at 4oC. At that temperature the density of water is 1 g/cm3, and the density of any substance divided by 1 g/cm3, will be numerically equal to that substance’s density, but will be unit-less; i.e. iron has a density of 7.85 g/cm3 and a specific gravity of 7.85

Figure 5 – Two objects with different densities but the same mass.

Density can be determined by the water displacement method or by using Archimedes’ principle. In the water displacement method, an object is placed into a graduated cylinder with a known volume of water. The object in the graduated cylinder will displace an equal volume of water to its own volume. By subtracting the original water level from the new water level, the volume of the object can be calculated. Archimedes’ principle states that a floating object displaces a mass of fluid equal to its own mass, and the mass of a submerged object is diminished by the mass of the displaced fluid. Thus, if we weigh an object in air and in water the following relationship can be used to determine its specific gravity or density:

Equation 2: obj obj

lost fluid

M M

 



Equation 3: air air water( )

M M M

  

Symbol Explanation Mobj Mass of object Mlost Difference in apparent mass of object in air and

water ρobj Density of object ρfluid Density of fluid Mair Mass of object in air Mwater Mass of object in water

Equation 2 takes into account that buoyancy appears to give an object a different weight. How easy is it to hold a person in water versus in air? The buoyant force of water helps suspend a person so they are easier to carry in water. Equation 3 shows how the density of objects can be determined by comparing the relationship of the weight of an object in water versus the weight of that object in air.

The quality of physics lab work depends mainly on how accurately the measuring tools are used. All measurements have some degree of uncertainty. For example, a watch is seldom exactly on time, as it may run a little fast or slow. A ruler is not perfectly accurate. It may be stamped imperfectly, or humidity and temperature may affect it by expanding or contracting the wood or metal and distorting the scale. Because uncertainty is unavoidable, a physicist must understand how uncertainty affects the outcome of an experiment. In this experiment, your hand, a meter tape, and a metric ruler will be used to make length measurements. A graduated cylinder will be used to measure volume via water displacement and a spring scale will be used to determine mass. A stopwatch will be used for basic time measurements. No measurement is complete without the units of measurement. Measurements and calculations must always include the units!

A. Experimental Error: Measured results will often differ from theoretical expectations, and there will be variations within repeated measurements. Use the Equations 4 and 5 to calculate these differences.

Equation 4: measured value accepted value %

Percent error accepted value

  

100

Equation 5: high value low value %

Percent difference average value

  

100

B. Significant Figures: The number of significant figures in a reading is dependent upon

the accuracy of the instrument. For instance, if a ruler can only measure to the millimeter length, such as 21.2 cm or 212 mm, the measurement cannot be written as 21.20 cm. While some may try to estimate the last number as 0, the number is only a guess and introduces uncertainty into the value. In this example, the digit 2 is the last significant figure reported. If the measurement corresponds exactly to a scale division, report the tenth of a centimeter as zero.

C. Average Deviation: To have confidence in experimental data, the measured quantity

must be able to be reproduced. Precision is a quantitative measure of the reproducibility of experimental measurements – that is, how well repeated measurements of the same quantity agree with one another. Precision is frequently measured in terms of the average deviation, which is determined by the following steps:

Average Deviation: 1. Determine the average value for the set of measurements; you must take at

least three measurements.

2. Determine the absolute value for the difference/deviation of each measurement from the average value.

3. Determine the average of the deviations.

Example: The density of a liquid is determined four times to be 1.64 g/mL, 1.60 g/mL, 1.59 g/mL, and 1.62 g/mL. The average density of the liquid is 1.61 g/mL. The individual deviations from the average value are 0.03 (1.61), 0.01 (1.60), 0.02 (1.59), and 0.01 (1.62). Therefore, the average deviation is 0.02. This deviation represents an uncertainty in the measurements. The density is not exactly 1.61 g/mL, but ranges from 1.59 g/mL to 1.63 g/mL. The measurement should be reported with the average deviation included: 1.61  0.02 g/mL. Note that an average measurement or an average deviation cannot have more decimal places than the numbers being averaged. An average deviation will not have as many significant figures as an average measurement.

Exercise 1: Estimation of Various Measurements

In this exercise, you will estimate the mass and length of different objects. You will then compare these estimates to actual measurements by calculating the percent error. You will also estimate and measure time to compare your estimates to the actual measurement and calculate the percent error. PROCEDURE:

1. Length

a. Prepare a table similar to Data Table 1 below to record observations while performing the experiment.

Data Table 1 — Estimation of various measurements

Measurement Estimated Actual % Error

Length (m)

Time (s)

Mass (g)

b. Estimate the length of a meter by putting a pen or pencil at one end of a table

and placing a second pen or pencil about one meter away from the first.

c. Use the meter tape measure to measure the actual length of your meter estimate.

d. Record the length of your meter estimate.

e. Calculate the percent error of your estimated meter from an actual meter.

2. Time:

1. Estimate a 30-s time period while someone else times you using a stopwatch. (If you do not have a partner, you can do this experiment by closing your eyes; start the stopwatch and stop it when you think 30 s have elapsed.)

2. Record the actual time of your estimate.

3. Calculate the percent error of the estimate from the actual time.

3. Mass:

a. Pick up a small paperback book or similar small object and estimate its mass.

b. Determine the actual mass of the object using the 500-g spring scale.

c. Record the estimated mass and the actual mass and calculate the percent

error.

Question: Why is it important to correctly estimate length, time, and mass?

Exercise 2: Measuring Using Instruments of Varying Degrees of Precision

In this exercise, you will use your hand, a ruler, and a tape measure to measure lengths of objects. You will then compare the precision of these different methods.

PROCEDURE:

1. Three objects will be measured in this exercise. Prepare three data tables similar to Data Table 2A, 2B, and 2C below to record observations while performing the experiment.

Data Table 2A — Measurement of an object using various instruments Length

(cm) Width (cm)

Height (cm)

Volume (cm3)

Object Being Measured:

Hand (hand units)

Hand (cm)

Ruler

Meter tape

Data Table 2B — Measurement of an object using various instruments Length

(cm) Width (cm)

Height (cm)

Volume (cm3)

Object Being Measured:

Hand (hand units)

Hand (cm)

Ruler

Meter tape

Data Table 2C — Measurement of an object using various instruments Length

(cm) Width (cm)

Height (cm)

Volume (cm3)

Object Being Measured:

Hand (hand units)

Hand (cm)

Ruler

Meter tape

2. Your hand:

a. Spread out your hand and use the measuring tape to measure the distance from

the tip of your thumb to the tip of your little finger in centimeters, and record the value.

Measurement of your hand span: ________ cm

b. Now use your hand to measure the length, width, and height of three rectangular

items such as small books, shoeboxes, or similar objects. The objects should weigh less than 500 g, so that you can also determine the mass if you wish.

c. Convert the hand units to centimeters and record in Data Table 2.

3. Metric ruler and meter tape:

a. Use the metric ruler to measure the length, width, and height of the same objects

from Step 1 and record the measurements in centimeters in Data Table 2. Be sure to place the markings on the ruler directly against the objects to minimize the possibility for error. Because the ends of rulers are often worn a bit, start your measurements at the 1-cm mark, then count the units rather than relying on the numbers marked on the ruler.

b. Record your measurements to the nearest half-millimeter. All your

measurements should have two places to the right of the decimal point and thus end with either a 5 or a 0, (for example, 12.35 cm or 9.60 cm).

c. Measure the length, width, and height of the box with the tape measure.

d. Record all measurements in Data Table 2. Units should be in centimeters and

recorded to the nearest half-millimeter as before.

4. Calculations: a. Find the volume of the object using the three different sets of measurements.

Remember, the volume of a rectangular box is: v = length × width × height. Show the units as cubic centimeters when recording the calculated volume in Data Table 2.

Questions:

A. Can you think of an occasion when it would be adequate to use your hand for measurement?

B. What would happen to your volume calculations if the length, width, and height

measurements were off a little?

Exercise 3: Graphing Data and Determining 

In this exercise, you will choose circular objects and measure the diameter and circumference of these objects. You will graph these measurements and calculate the slope. Then you will compare the value of the slope to the true value of , by calculating the percent error. PROCEDURE:

1. Prepare a table similar to Data Table 3 below to record observations while performing the experiment.

Data Table 3 — Determination of 

Object Diameter D

(cm)

Circumference C

(cm)

Slope % Error

2. Select five circular objects of different sizes, such as an AAA battery, a screw cap

from a soft drink bottle, the cardboard center of a paper towel roll, cups of various sizes, or plates of various sizes.

3. Use the metric ruler or tape measure to measure the diameter D in centimeters of

each object to two decimal points and record in Data Table 3.

4. Using the tape measure, measure the circumference C in centimeters of each object to two decimal points and record in Data Table 3.

5. Graph C vs. D using a computer spreadsheet program.

6. Use the linear fit command from the program menu to plot a best-fit line. Remember,

the equation for the slope of the line is y = mx + b, where the slope is m.

7. What is the slope of the line? What does it represent? Record the slope of the line for each object in Data Table 3.

8. Calculate the percent error of this value obtained from the slope from the true value

of  and record in Data Table 3.

NOTE: Circumference is 2r and diameter is 2r

Exercise 4: Density Measurements

In this exercise, you will use two different methods to measure the density of a metal bolt: water displacement and Archimedes’ principle. You will then compare the accuracy of these two methods. PROCEDURE:

1. Determine the density of a metal bolt (or any irregular metal object) by the water- displacement method:

a. Prepare a table similar to Data Table 4 below to record observations while

performing the experiment. Data Table 4 — Density measurements Method Volume of

water in graduated

cylinder (mL)

Volume of water + bolt (mL)

Volume of bolt (mL)

Mass of bolt in air

(g)

Mass of bolt in water

(g)

Mass of bolt

“lost” in water

(g)

Density or

S.G. of bolt

(g/mL) S.G.=

unitless Water-

displacement method

Archimedes’ principle method

b. Tie a string around the metal bolt and attach the string to the bottom of the 10-g spring scale so that the bolt hangs down about 5 cm. Record the bolt’s mass in air in Data Table 4. Place this value into both rows for “water- displacement method” and “Archimedes’ principle.”

c. Half-fill the graduated cylinder and record the volume of the water without the

object. Record this volume in Data Table 4.

d. Place the metal bolt into the graduated cylinder and record the new volume (water + bolt) in Data Table 4. The difference between the two volumes represents the volume of the bolt (or other object). Record the difference (volume of bolt) in Data Table 4.

e. Calculate the density of the bolt by dividing the mass in air by the change in volume, and record in Data Table 4.

2. Determine the specific gravity or density using Archimedes’ principle:

1. Partially fill a cup with water. The cup needs to be large enough to suspend a

metal bolt.

2. While holding the top of the 10-g spring scale, suspend the metal bolt hanging from a string into the partially filled cup of water. Make sure that the bolt does not touch the sides or bottom of the cup.

3. Read the 10-g spring scale. This value is the bolt’s mass in water. Record it in Data Table 4.

4. Subtract the bolt’s mass in water from the bolt’s mass in air. This difference is the apparent mass lost in water. Record this new value in Data Table 4.

5. To calculate specific gravity , divide the bolt’s mass in air by the bolt’s apparent mass lost. Record the specific gravity of the bolt in Data Table 4. To convert specific gravity to density, simply add g/cm3 to the specific gravity values.

Questions:

A. Which of the two density or specific gravity determinations will be more accurate? Explain your answer.

B. Research the Archimedes’ principle method. Write one paragraph explaining why it is called Archimedes’ principle.

Exercise 5: Time Measurements

In this exercise, you will drop an object and measure the time that it takes to fall to the floor. In the first part of the exercise, you will use visual clues to measure the time it takes for the object to hit the floor. In the second part of the exercise, you will use auditory clues to measure the same time interval. Then you will compare the results of these measurements. PROCEDURE:

a. Prepare a table similar to Data Table 5 below to record observations while performing the experiment. Data Table 5 — Time measurements using visual cues

Drop time (s) Trial 1 Trial 2 Trial 3 Average

b. Measure and mark a vertical distance of 2 m from the floor up.

c. Stand on a chair and hold a small box or similar object (such as a marble) at the

marked height in one hand and the stopwatch in the other hand.

d. Start your stopwatch at the same instant you release the object and stop the timer when you see the object hit the floor.

e. Record the time to the nearest tenth of a second in Data Table 5.

f. Repeat three times.

TIP: If you have an assistant, have them time you while you drop the box – use verbal commands like “start” or “now” to synchronize the dropping and timing.

g. Find the average drop time of the object and record it in Data Table 5.

h. Prepare a table similar to Data Table 6 below to record observations while performing

the experiment. Data Table 6 — Time measurements using auditory cues

Drop time (s) Trial 1 Trial 2 Trial 3 Average

i. Repeat the object-dropping steps with your eyes closed. Stop the stopwatch when you

hear the object hit the floor. Record all the data (including the calculated average) in Data Table 6.

Questions:

A. Which is more accurate, the individual times or the average? Explain your answer.

B. Sometimes when many trials are run and recorded, the highest and lowest data points are disregarded when taking the average. Would this technique work in this experiment? Explain your answer.

C. Explain any differences that occurred in your results between recording the data visually and aurally.

Trigonometric Measurements

Experiment Summary:

Students will have the opportunity to review basic trigonometric functions. They will use trigonometry to calculate the height of a building using a weight hanging from a protractor. Then they will calculate the degree of accuracy of

the calculations.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To review basic trigonometric functions, and  To measure the height of a building using trigonometry.

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 1 Masking or Scotch® tape 1 Computer and spreadsheet program 1 A lab partner (optional) From LabPaq 1 Protractor 1 Straw, Drinking 1 Tape measure, 1.5-m 1 Tape measure, 3-m String & Weight Bag String & Weight Bag 1 String - Qty-4.0 Meters 1 Weight, 1/2 oz.

Discussion and Review

A review of trigonometry is required for this experiment. A. Determination of the sides of a right triangle when the hypotenuse and one angle are measured: Using the triangle in Figure 1, the length of the hypotenuse, C, is measured with a meter tape or ruler. Angle B is then measured with a protractor. Then the lengths of the other two sides of the triangle can be determined by trigonometry. To determine the length of side b, when the hypotenuse and angle B are known, use is made of the sine function defined as:

sinB = b/c Solving for side b we obtain: b = c sinB

B. Determination of the hypotenuse and angles of a right triangle when the sides are measured: If sides a and b of a right triangle are measured the hypotenuse, C, is obtained by the Pythagorean Theorem. That is, c2 = a2 + b2 and c = √ (a2 + b2) Angle B of the triangle is obtained by the use of the tangent function. That is, tan B = b/a Solving for B by taking the inverse of the above equation:

B = tan -1 b/a Figure 1: Right Triangle C. Indirect measurements of heights by trigonometry: Trigonometric functions can be used to make an indirect measurement of a height. As an example, suppose you wanted to determine the height of a building. However, you cannot directly measure it with a tape measure because the building is too high. Yet the height can easily be determined by modifying a protractor. Tape an ordinary soda straw along the straight edge of the protractor as shown in Figure 2. The straw is used as a sighting instrument. A hole is drilled through the center mark of the protractor and a short length of string with a weight is hung from the center mark to hang below the protractor’s edge.

Figure 2: Modified Protractor Fitted with a String and Hanging Weight and Its Use

The weighted string acts as a plum bob and determines the location of the vertical. When the straw is horizontal the string should cross the 90o mark on the protractor. To look through the straw to the top of the building the protractor must be rotated. The string will read the angle that the protractor has rotated through, and as can be seen in Figure 2b above this is the elevation angle theta of the building. If the distance from the student to the base of the building is measured as "a" then side "b" of the triangle is the height of the building and can be found from the tangent functions: tan θ = b/a and B = a tan θ Now b is not quite the height of the building because the distance from the ground to the location of the student's eyes, ho, must also be taken into account. Taking this correction into account, the height of the building is H = b + ho PROCEDURE: 1. Experimental measurement of the angles and sides of a right triangle:

a. Create a triangle by taping a string against a wall and taping the bottom of the string to the floor or a table set against the wall. Make sure that the wall is perpendicular to the floor or table by measuring angle C, which should be 90o.

b. Measure sides a, b, and c of the triangle with a tape measure and record these

values in centimeters (cm) on your data sheet.

c. Measure angles A and B of the triangle with a protractor and record on your data sheet.

d. Using the values for the sides of the triangle that you measured in Step B, compute

angles A and B by using the different trigonometric functions.

e. Calculate the average value of the computed angle A by adding the three computed values of A and dividing by 3. Record this value on your data sheet.

f. Calculate the average value of the computed angle B by adding the three computed

values of B and dividing by 3. Record this value on your data sheet.

g. Compare the values of the measured angles with the average computed values and determine their difference.

2. Determination of the sides of a right triangle when the hypotenuse and one angle are

measured:

a. Create a new triangle by taping the bottom of the string to a different position on the floor or table, as in Procedure 1.

b. Measure the hypotenuse c of the triangle with a tape measure and record this value

in centimeters (cm) on your data sheet.

c. Measure angle B with a protractor and record on your data sheet.

d. Calculate sides a and b of the triangle by trigonometry and record these values on your data sheet.

e. Measure sides a and b of the triangle with a tape measure and record these values

in centimeters (cm) on your data sheet.

f. Determine the difference between the measured and calculated values of the sides of the triangle and record on your data sheet.

3. Determination of the hypotenuse and angles of a right triangle when the sides are

measured: a. Create a new triangle by taping the bottom of the string to a different position on the

floor or table, as in Procedure 1.

b. Measure sides a and b of the triangle with a tape measure and record these values in centimeters on your data sheet.

c. Calculate the hypotenuse c of the triangle by using the Pythagorean theorem and record on your data sheet.

d. Measure the hypotenuse c of the triangle with a tape measure and record this value

in centimeters on your data sheet.

e. Calculate the difference between the measured and calculated values of the hypotenuse and record on your data sheet.

f. Measure angle B of the triangle by using a protractor and record on your data sheet.

g. Calculate angle B of the triangle by using the tangent function and record on your data sheet.

h. Calculate the difference between the measured and calculated values of angle B and

record on your data sheet.

i. Since the sum of the three angles of a triangle is equal to 180o, calculate angle A of the triangle.

4. Indirect measurements of heights by

trigonometry:

a. Prepare to measure the height of any tall building on campus or in your neighborhood. This is best done with a partner. You will use a protractor fitted with a drinking straw and weighted string as previously shown in Figure 2. Don’t forget to describe the building in your report.

b. Move some distance away from the building (at least 5 to 10 meters or if possible, a

distance equal to the height of the building). Sight through the straw to a point on the top of the building. When the top is sighted the second person should read the elevation angle “theta” from the protractor. Record the elevation angle “theta” on your data sheet.

c. Using the tape measure, measure the distance “a” in meters from the sighting

position to the wall of the building and record this value on your data sheet.

d. Measure the vertical distance ho from the ground to your eyes in meters and record this value on your data sheet.

e. Calculate the height b and record on your data sheet.

f. Calculate the height h of the record value in meters on your data sheet.

g. Repeat Steps b through f for two different a distances (not less than 5 meters) from

the building.

h. Using the data from Step g calculate the average height h of the building.

Questions:

A. Draw a triangle with sides measuring 8, 12, and 14.4 units. With a protractor measure the angles. Then compute the angles trigonometrically.

B. What would happen in your previous experiments if the wall or building were not

perpendicular to the ground?

C. What uncertainty is introduced into the experiment by using a tape measure to measure the sides of the triangle?

D. To what degree of accuracy can you read a protractor?

Data Collection

Experiment Summary:

Students will form a hypothesis for the pitching velocity of a ball. Then they

will use math and spreadsheets to calculate the actual velocity and determine the accuracy of their hypothesis. Students will also roll a large ball

to measure its velocity and graph its horizontal motion. *Modified from P.W. Laws, 1997

Peter Jeschofnig, Ph.D. Version 09.1.02

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set aside a safe

workspace in which to complete the exercise.

Objectives

1. To form a hypothesis about velocity and then collect velocity data.

2. To measure the velocity of a pitched ball.

3. To make a hypothesis about the motion of a ball on a horizontal surface.

4. To study the mathematical relationships of velocity and how these

relationships are represented in graph form.

Estimated time to perform this experiment: 3 hours

Materials

Materials: Qty Item Description

Student Provides 1 Tennis ball, baseball, or similar ball 1 Catcher’s mitt (optional) 1 Computer and spreadsheet program 1 Masking tape for marking distances 1 Bowling ball (ideal) or other large ball (soccer ball,

basketball, beach ball, etc.) 2 Lab partners 1

Large open space in which to pitch and roll balls Cardboard, board or other stiff object for a ramp Book to stack for ramp

LabPaq Provides 1 Stopwatch-digital 1 Tape measure, 3-m

Discussion and Review

Ibn al-Haytham is considered one of the founding fathers of the scientific method and made significant contributions in physics, mainly in the field of optics. Around 1000 B.C., he stated: “Truth is sought for its own sake. And those who are engaged upon the quest

for anything for its own sake are not interested in other things. Finding the truth is difficult, and the road to it is rough.”

The scientific method is a set of techniques used by the scientific community. This method is used to investigate natural phenomena by providing an objective framework in which to make scientific inquiry. Scientists analyze data so that a conclusion can be reached about that inquiry.

The goal of the scientific method is to find answers to questions. The method has a series of steps but the order in which these steps are taken sometimes vary. The scientific method is not a recipe, but rather an ongoing cycle that is applied with intelligence, imagination, and creativity. Frequently, some of these steps will take place simultaneously, or in a different order, or will be repeated as the experiment is refined. The following is the most general and intuitive sequence for the scientific method:

1. Ask a question – A scientist asks questions about a natural phenomenon or group of phenomena that piques their curiosity. This is the first step in finding an explanation about the subject that interests them.

2. Research the topic – In order to formulate a hypothesis, some research has to be

conducted in order to better understand the topic. This increase in knowledge will assist the scientist in developing a hypothesis that is relevant to the topic.

3. Formulate a hypothesis – Research on the topic is used to generate a hypothesis, an

educated guess, about what will be learned in the experiment.

4. Test the hypothesis – The scientist then plans and implements a procedure that will test the hypothesis. The experiment has to include quantifiable data that can be analyzed.

5. Analyze the data –The scientist uses proper mathematical analysis to see if the

results of the experiment support or refute the hypothesis. If the data do not support the hypothesis, the scientist must reject, modify, and re-test the hypothesis. Frequently, the results of the experiment are compiled in the form of a lab report (for classroom work), or a published paper (for academic research). The results of the experiment frequently provide an opportunity for more questions about the phenomena being studied, and so the process of inquiry is repeated with new questions.

Exercise 1: Formulating a Hypothesis about Baseball Pitching Speed

In professional sports, radar is used to measure pitching speeds. A radio transmitter sends out a radio signal, which bounces off the moving object. The frequency of the radio wave changes due to the Doppler Effect and this frequency is picked up by the receiver. The

receiver uses this difference in frequency to calculate the speed of the object.

In this experiment, you will make a hypothesis about your pitching speed. This hypothesis will be compared to the actual speed that will be measured. This exercise is designed to provide you with experience in making hypotheses and then taking measurements to test hypotheses. You will be using both hand calculations and spreadsheet calculations to find speeds based on measurements of times and distances. These experiments require two partners, so recruit friends or relatives to work with you for a short time.

Suppose you were to go outside, stand at a comfortable distance from a lab partner, and pitch a ball to him or her. How fast do you think you can pitch? Note: The distance from a pitcher's mound to home plate is 18.45 meters (m) or 60.5 feet (ft). A world-class professional pitcher can throw a baseball at a speed of just over 2683.3 meters per second (m/s) or 100 miles/hour (mi/h). See Figure 1.

PROCEDURE: 1. Hypothesizing on your pitching speed:

1. Prepare a data table similar to Data Table 1 shown below. Data Table 1– Pitching speeds

Hypothesis for pitching speed: Name: ______________ (meters/second): _______________

A B C D E F G = B/F H

Pitcher Name

Distance (m)

Time t1 (s)

Time t2 (s)

Time t3 (s)

Avg. time (s)

Avg. speed (m/s)

Avg. speed (mi/h)

2. Make a hypothesis for the speed that you think you can comfortably pitch in miles

per hour. Convert units to meters per second. There are 1,609.344 m in 1 mi. 3. Explain how you derived this hypothesis. Record it in Data Table 1.

4. Would you call your hypothesis a guess, a prediction, or something else? Explain

your answer. Record it in your lab report. 2. Collecting data on pitching speeds:

a. You will need two partners for this activity. Find an open area in which to throw a ball and catch it. This area needs to be at least 20 m (66 ft.) in length.

b. At one end of the area, the pitcher will stand and throw the ball. At the other end of the area, a second person will stand to catch the ball. Mark off a distance between these two people ranging from 5 to 15 m. If the pitcher is a good thrower, this distance can be extended to 20 m or more depending on their capability. Record the distance between the catcher and pitcher in Data Table 1, column B.

c. A third person will need to stand off to the side, but in an area between the pitcher and catcher so they can time the ball speed.

d. Devise a system for timing the pitches. An example is to have the pitcher yell “throw” so the person timing can start the timer, and have the catcher yell “caught” so the person timing can know when to stop the timer.

e. To avoid injury, have the pitcher warm up and stretch properly before pitching.

f. When everyone is in position, have the pitcher throw three pitches while the timer records the time it takes the ball to travel to the catcher. Record these times in seconds in Data Table 1, columns C, D, and E.

g. Have three different people pitch, and then enter your data into Data Table 1. As an

alternative, you can exchange data with online classmates through discussion groups.

h. Calculate the average time for each person and enter into Data Table 1, column F.

i. Use the data from columns B and F to calculate the average speed in both meters per second and miles per hour. Average speed is distance divided by time. Record these values into in Data Table 1, columns G and H. Use three significant figures in your answer.

j. Compare your throwing results to your hypothesis. How well did you hypothesize?

Calculations with computers and spreadsheets: Spreadsheets are invaluable tools for data analysis. A computer spreadsheet is used to calculate data. When entering formulas into a spreadsheet, the program performs the calculations. If a mistake is made, the correct number can be re-entered and the spreadsheet program recalculates everything automatically. Physicists often display and perform calculations on spreadsheets using data tables like the one used for this exercise.

1. Open a spreadsheet program such as Microsoft Excel® on your computer.

If you are not already familiar with its use, take some time to review the tutorial located in the lab manual.

2. Enter all data from Data Table 1 into a spreadsheet.

3. Enter formulas into the spreadsheet to calculate the average times and

speeds of the baseball for each student and for the group as a whole.

4. Save your spreadsheet work into a computer file so it can be reloaded later. Note: If your spreadsheet cells contain more than three significant figures use the "format cell" feature to reduce the number of significant figures.

How do the average times and speeds calculated by your spreadsheet program compare to the manual calculations you made on the same data in Exercise 1? If your calculations differ, explain why this may have occurred.

Exercise 2: Hypothesizing about and Measuring the Motion of a Ball on a Horizontal Surface

A key to understanding how to describe motion near Earth’s surface is to observe horizontal motions and vertical motions separately. Eventually, situations in which an object undergoes both horizontal and vertical motion can be analyzed and understood as a combination of these two kinds of basic motions. See Figure 2. In this exercise you will make hypotheses about the horizontal motion of a bowling ball (or other large ball).

Figure 2 – Forces on an object moving on a horizontal surface. If these forces have a net value of zero, a moving object moves at a constant velocity.

Item Description 1 Force of ground on object 2 Force of object moving forward while pushed or pulled 3 Force of gravity on the object 4 Force of friction on the object while pushed or pulled

PROCEDURE:

1. Prepare a data table similar to Data Table 2 shown below.

Data Table 2 – Rolling a ball along a horizontal surface Time (s):

2-m Distance Time (s):

4-m Distance Time (s):

6-m Distance Trial 1 Trial 2 Trial 3 Average time (s) Avg. velocity(m/s)

2. Visualize the horizontal motion of the ball in Figure 2. If you roll a ball along a

surface, will the velocity stay the same as it rolls? Make a hypothesis about the velocity of the ball as a function of time. Record this hypothesis in your lab report. Hypothesis:

NOTE: This experiment should be conducted in a very long room, a parking lot, or a sidewalk. Chose a place to roll the ball that has a smooth level surface.

3. Use masking tape to mark off 6 m (19.685 ft.) in 1-m (39.37-in) increments along the

surface you have chosen to roll the ball. Use the first marker for the starting line.

4. Select two lab partners to assist with timing and recording the time during this experiment. One person will be the timer while the other person will watch the ball and keep track of the times. One person will roll the ball. The watcher will say the distance aloud when the ball passes each mark. Then the timer will announce the time the mark was passed, in seconds, so it can be recorded. This process will continue until the ball has passed the 6-m mark.

5. Roll the ball along the surface so it passes the 6-m mark. Observe and record the

times it takes to pass the following marked distances: 2.0 m, 4.0 m, and 6.0 m. Start the timer when the ball begins to roll and note the times when the ball passes each mark. Because you are measuring the time between each interval in one roll, subtract the time at the 4-m mark from the time at the 2-m mark. Do likewise for the 6-m mark, subtracting the time at the 4-m mark. An example of how this is calculated is shown below:

Examples:  2-m: 0.5 s: this is 0.5 s/2 m = 0.25m/s  4-m: 1.2 s: 1.2 – 0.5 = 0.7 s/ 2 m = 0.35m/s  6-m: 2.0 s: 2 - 1.2 = 0.8 s/2 m = 0.4m/s

6. Record the calculated time results, for each distance, in Data Table 2 for Trial 1.

7. Repeat this experiment for three trials and record all times for each distance in Data Table 2. NOTE: Try to use the same force each time the ball is rolled.

8. For each distance, add up the three trials and divide by 3 to find the average. Record

these average values in Data Table 2.

9. Calculate the average velocity of each distance and record in Data Table 2. Average velocity is calculated by dividing the distance by the average time. An example of how to do this calculation is shown in Step 4.

10. Open a computer spreadsheet and transfer the data from Data Table 2 into it.

11. Graph the data for the distance the bowling ball traveled as a function of the rolling- time of the ball. Your computer graphs (one for each trail) should have distance in meters on the y-axis versus time in seconds on the x-axis.

Exercise 3: Examining how the Ball’s Distance Varies with Time

In this exercise, the student will examine how different mathematical relationships are represented in graphs. PROCEDURE:

1. Look at the examples of mathematical relationships between distance and time for a ball rolling on a level surface in Figures 3, 4, and 5.

2. Compare these relationships to your graph and then answer the following questions.

Do any of the following graphs match the graph you prepared in Exercise 2?

Figure 3 – Mathematical relationship: y is directly proportional to x. The equation for the

slope of a line is y = mx + b.

Figure 4 – Mathematical relationship: As y (distance) increases, x (time) increases.

Figure 5 – Mathematical relationship: The equation for this straight line is y = mx +b, like in

Figure 3, but the line does not go through the origin, and the y-intercept is not 0.

QUESTIONS

A. Compare the shapes of the graphs you produced in Exercise 2 with the graphs shown in Figures 3, 4, and 5. Do the graphs you produced fit any of the functions illustrated in these figures? Explain your answer.

B. How did the movement of the ball compare to the hypothesis you made in Exercise 2,

Step 1?

C. Refer to the graphs of time versus distance you made for the rolling ball. What would happen to the slope m if the ball rolled faster?

D. What sources of error were involved in this experiment and how could the amount of error be reduced?

Exercise 4: Vertical Acceleration Due to Gravity

In this exercise, you will explore the motion of falling objects. As an object falls, the velocity of the object will increase due to the acceleration caused by Earth’s gravity. A stopwatch will be used to determine the time it takes for a dropped object to fall from a certain height to the floor. From this data, you will calculate the gravitational acceleration.

In this experiment, you will use the equation shown below to calculate the gravitational acceleration.

For an object that starts from rest and then accelerates at a constant rate g, the distance d it travels after time t is expressed as:

d = ½ gt2

rearranged to solve for g as : g = 2d/t2

The value for g (gravitation acceleration) is 9.81 m/s2. However, the value calculated in this experiment may be different due to experimental error. The reaction time of starting and stopping the timer and noting the exact time that the ball has reached the floor will influence the data collected in this exercise.

PROCEDURE:

1. Prepare a data table similar to Data Table 3 shown below.

2. Hypothesize an experimental value of g obtainable through this simple stopwatch timing experiment. Hypothesis on the experimental value of g: _________m/s2

Data Table 3 – Time trials to drop an object 2 m

Item Data Trial 1 Drop time, t, (s) Trial 2 Drop time, t, (s) Trial 3 Drop time, t, (s) Trial 4 Drop time, t, (s) Trial 5 Drop time, t, (s) Trial 6 Drop time, t, (s) Trial 7 Drop time, t, (s) Trial 8 Drop time, t, (s) Trial 9 Drop time, t, (s)

Trial 10 Drop time, t, (s) Average drop time (s)

t2 (s2) Experimental

g = 2d/t2 (m/s2)

3. Use the tape measure to measure and mark a vertical distance of 2 m from the floor.

4. Stand on a solid chair and hold a small object, like a marble, at the marked height. Have a partner use a stopwatch to record the time it takes for the object to fall 2 m.

5. Tell your partner when to start timing as you drop the ball. Have the timer watch when the ball hits the floor so they can stop timing.

6. Record this time as “Trial 1 Drop time, t, (s)” in Data Table 3. Record to the nearest tenth of a second.

7. Conduct Trials 2-10 by dropping the object and recording the times in Data Table 3. 8. Calculate the average drop time of the object, and record in Data Table 3.

9. Take the square of this average time and record in Data Table 3, row “t2 (s2)”.

10. Calculate g using the equation g = 2d/t2 and record it into Data Table 3.

11. Compare the calculated magnitude of acceleration due to gravity to your hypothesis

made in Step 1. How close was your hypothesis to the results obtained in this exercise?

Questions A. Calculate the % error between the experimental g recorded in Data Table 3 and the

theoretical value of 9.81m/s2.

% error = experimental value – theoretical value × 100 theoretical value

B. What are some possible errors in the design of this lab?

C. What improvements could be made to reduce error?

Acceleration

Experiment Summary:

Students will have the opportunity to explore Galileo’s studies on the rates of objects rolling down inclines. They will look at both the horizontal and vertical

forces acting on a ball as it rolls down an incline. They will also measure a marble’s velocity as it rolls down an incline and then calculate the rate of

acceleration.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Materials

 To calculate the acceleration of an object rolling down an inclined plane.

Materials

Materials From: Label or Box/Bag: Qty Item Description: Student Provides 1 Ramp about 1 – 2 meters long

made of wood, grooved trim molding, or stiff cardboard

From LabPaq 1 Stopwatch-digital 1 Tape measure, 3-m Marble, Bolt, & Spring

Bag 1 Marble

Discussion and Review

According to the ancient doctrine of Aristotle, every element seeks its own level. Thus stones fall toward the center of the earth, water flows downhill, and air and fire rise. Ancient writers maintained that each object would fall at a fixed speed which depends on the amount of the Element Earth it contains, and thus on its weight. This doctrine was still being taught at the University of Pisa around 1600 when it was challenged by Galileo, then a junior professor. Galileo argued that if the effects of air and other resistances are removed all objects fall at the same rate. However, the speed is not constant as taught by Aristotle but increases continually as long as the body falls. It is the rate of change, the acceleration which is constant, not the velocity. Galileo was a pioneer of the modern scientific method and advocated sound logical argument based on careful physical observation. Although he probably did not perform the legendary demonstration from the Leaning Tower of Pisa as alleged by his biographer Viviani (though a similar demonstration was done by Simon Stevin in Bruges a few years earlier), he made a very effective experiment using an inclined plane. His arguments in his later work The Two Great Systems of the World (1632) show that he had explicitly realized that acceleration exists in the vertical dimension only, that horizontal motion is independent of acceleration, that an oblique constraint such as an inclined plane equilibrates part of the acceleration, and that acceleration is therefore proportional to what today we would call the sine of the angle of inclination. Galileo showed mathematically that if the acceleration is constant then the distance traveled by a ball rolling down such a plane would be proportional to the square of the elapsed time (that is x = ½ at2). Then he demonstrated that the distance actually covered by a rolling ball has exactly this relation to the time. Modern clocks had not yet been invented so Galileo measured time by his own pulse or by a water clock. He showed that acceleration which is constant for a given angle of inclination increased as the angle was steepened until the motion was too fast to be timed accurately. However, he argued that the data could be extrapolated until the plane was inclined vertically, whereupon the ball would be in free fall. There are many variations on Galileo’s original demonstration including one in which a ball is projected horizontally across an inclined board. The path is traced and shown to be a parabolic trajectory, which implies a constant acceleration in the direction of inclination. Other methods employ a ball or low-friction cart on an inclined path or a trolley on an inclined wire with direct timing by stopwatch, photogate, or spark track. Each method has its own peculiar difficulties with the accuracy and precision of timing, straightness of the track (and thus variation in the angle of inclination), friction, and air resistance. Any rolling parts, such as the ball in Galileo’s original demonstration, not only present rolling friction but also divert part of the linear acceleration into angular acceleration proportional to the angular momentum of the parts. Acceleration is a change in the velocity of an object. Velocity is a vector quantity with both direction and magnitude. Acceleration is also a vector quantity having both direction and magnitude.

If the speed of an object is changed that object has accelerated either positively or negatively, depending on whether it increased or decreased speed. Another way to accelerate an object is to change its direction of movement. That means a car going around a corner is undergoing acceleration because its velocity in terms of direction is changing even if the car’s speed as seen on the speedometer is constant. As discussed above, an object falling under the influence of gravity accelerates. You know from your studies the four kinematic equations for the uniformly accelerated motion of an object starting from rest; v=velocity, a=acceleration, and x=distance.

Equation 1: v = at Equation 2: x = (v/2)t

Equation 3: v2 = 2ax Equation 4: x = 1/2at2

Exercise 1

50 cm 100 cm

150 cm

PROCEDURE:

1. Set up a ramp as shown in Figure 1. Depending on the length of the board mark off three distances along the ramp, such as 30 cm, 60 cm and 90 cm or 50 cm, 100 cm, and 150 cm. The wider the spacing on a longer board may give slightly better data. Prepare data and calculation sheets similar to the one that follows.

2. Measure the height of the ramp and record.

3. Measure the angle of the inclined ramp using a protractor. You may use the

protractor with plumb line you used in the previous trigonometry experiment (see Figure 2). Figure 1. Figure 2.

a. Hold the marble at the top of the ramp.

b. Release the marble and start the stopwatch simultaneously.

c. As the marble passes the first marked distance, stop the watch.

d. Record the time to the nearest tenth second in Table 1.

e. Repeat the procedure at least four times.

f. Repeat timing to the second marked distance, third distance, etc.

DATA TABLE 1: Height of ramp: _________m; Angle of incline = ____ o TRIAL No. Distance (x) –

m Time (t) - seconds

Velocity (v) – m/s

Acceleration (a) – m/s2

1 .30 2 .30 3 .30 4 .30 5 .30 Average = Average = Average = 6 .60 7 .60 8 .60 9 .60 10 .60 Average = Average = Average = 11 .90 12 .90 13 .90 14 .90 15 .90 Average = Average = Average =

Calculations: Notice that your data consists of times (t) and distances (x). In order to calculate acceleration you must use the four equations of motion. First work with Equation 2:

Equation 2: x = (v/2)t

Rewriting to isolate v, the equation becomes:

Equation 2: v= (2x)/t You already know the distance, x and the time, t. Now you can solve for velocity, v, by using Equation 2 in this form. Fill in v for each trial on your data sheet. Use Equation 1 to find acceleration, but first it must be rewritten to isolate the variable (a).

Equation 1: v=at divide both sides of the equation by (t): a= v/t

With Equation 1 rewritten in this way it is easy to solve for acceleration, a using time, t from your data sheet and velocity, v that you already calculated. Fill in the value for acceleration in each trial on your calculation table.

Questions:

A. Newton’s first law says a body at rest will remain at rest unless acted upon by an outside force, and a body in motion will continue in motion at the same speed and in the same direction unless acted upon by an outside force. What forces were acting on the marble as it traveled down the ramp?

B. Did the velocity of the marble increase as it traveled down the ramp? C. Did the acceleration of the marble increase as it traveled down the ramp? D. What would happen to the velocity and acceleration of the marble if the ramp were

steeper? How about if the ramp were vertical? If you have trouble answering this, repeat the experiment with a steeper ramp.

E. As you are riding in a car your body is at rest relative to the car but in motion relative

to the street. What might happen to the passengers in a car during a sudden stop or crash if there are loose objects stacked in the rear window of the car?

Friction

Experiment Summary:

Students will have the opportunity to explore the concept of friction. Using a spring scale, students will pull wood blocks with varying surface compositions

to calculate the coefficient of friction. Students will calculate the forces needed for both static and kinetic friction to move the block from a resting

position and from a position of movement. Students will pull the block across varying surfaces to observe how different frictional forces affect the value of

the friction coefficient.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To provide an understanding of the concept of friction.  To calculate the coefficient of friction of an object by two methods.

Materials

Materials From: Qty Item Description:

Student Provides 1 Ramp board: 3 - 4 feet long, 10 cm wide 1 Can of soft drink or item of similar weight LabPaq 1 Friction block set-PK 1 Protractor 1 Scale-Spring-500-g 1 Tape measure, 1.5-m 1 Tape measure, 3-m

Discussion and Review

Whenever a body slides along another body a resisting force is called into play that is known as friction. This is a very important force and serves many useful purposes. A person could not walk without friction, nor could a car propel itself along a highway without the friction between the tires and the road surface. On the other hand, friction is very wasteful. It reduces the efficiency of machines because work must be done to overcome it and this en- ergy is wasted as heat. The purpose of this experiment is to study the laws of friction and to determine the coefficient of friction between two surfaces. THEORY: Friction is the resisting force encountered when one surface slides over another. This force acts along the tangent to the surfaces in contact. The force necessary to overcome friction depends on the nature of the materials in contact, on their roughness or smoothness, and on the normal force but not on the area of contact or on the speed of the motion. We find experimentally that the force of friction is directly proportional to the “normal force.” When an object is sitting on a horizontal surface the normal force is just the weight of the object. However, if the object is on an incline then it is not equal to the weight but is calculated by N= mg cos θ. The constant of proportionality is called the coefficient of friction, μ. When the contacting surfaces are actually sliding one over the other the force of friction is given by Equation 1: Ffr = k FN where Ffr is the force of friction and is directed parallel to the surfaces and opposite to the direction of motion. FN is the normal force and k is the coefficient of kinetic friction. The subscript k stands for kinetic, meaning that k is the coefficient that applies when the surfaces are moving one with respect to the other. k is therefore more precisely called the coefficient of kinetic or sliding friction. Note carefully that Ffr is always directed opposite to the direction of motion. This means that if you reverse the direction of sliding the frictional force reverses too. In short, friction is always against you. Friction is called a “non- conservative” force because energy must be used to overcome it no matter which way you go. This is in contrast to what is called a “conservative” force such as gravity, which is against you on the way up but with you on the way down. Thus, the energy expended in lift- ing an object may be regained when the object descends. Yet, the energy used to overcome friction is dissipated, which means it is lost or made unavailable as heat. As you will see in your later study of physics the distinction between conservative and non-conservative forces is a very important one that is fundamental to our concepts of heat and energy. A method of checking the proportionality of Ffr, and FN and of determining the proportionality constant k is to have one of the surfaces in the form of a plane placed horizontally with a pulley fastened at one end. The other surface is the bottom face of a block that rests on the plane and to which is attached a weighted cord that passes over the pulley. The weights are varied until the block moves at constant speed after having been started with a slight push. Since there is no acceleration the net force on the block is zero, which means that the frictional force is equal to the tension in the cord. This tension, in turn, is equal to the total weight attached to the cord's end. The normal force between the two surfaces is equal to the weight of the block and can be increased by placing weights on top of the block. Thus, corresponding values of Ffr, and FN can be found and plotting them will show whether Ffr and FN are indeed proportional. The slope of this graph gives k.

When a body lies at rest on a surface and an attempt is made to push it the pushing force is opposed by a frictional force. As long as the pushing force is not strong enough to start the body moving the body remains in equilibrium. This means that the frictional force automatically adjusts itself to be equal to the pushing force and thus to just be enough to balance it. However, there is a threshold value of the pushing force beyond which larger values will cause the body to break away and slide. We conclude that in the static case where a body is at rest the frictional force automatically adjusts itself to keep the body at rest up to a certain maximum. But if static equilibrium demands a frictional force larger than this maximum, static equilibrium conditions will cease to exist because this force is not available and the body will start to move. This situation may be expressed in equation form as: Equation 2: Ffr ≤ sFN or Ffr max = sFN Where Ffr is the frictional force in the static case, Ffr max is the maximum value this force can assume and s is the coefficient of static friction. We find that s is slightly larger than k This means that a somewhat larger force is needed to break a body away and start it sliding than is needed to keep it sliding at constant speed once it is in motion. This is why a slight push is necessary to get the block started for the measurement of k.

One way of investigating the case of static friction is to observe the so-called “limiting angle of repose.” This is defined as the maximum angle to which an inclined plane may be tipped before a block placed on the plane just starts to slide. The arrangement is illustrated in Figure 1 above. The block has weight W whose component Wcos  (where  is the plane angle) is perpendicular to the plane and is thus equal to the normal force, FN. The component W sin  is parallel to the plane and constitutes the force urging the block to slide down the plane. It is opposed by the frictional force Ffr, As long as the block remains at rest, Ffr must be equal to W sin . If the plane is tipped up until at some value max the block just starts to slide, we have:

Equation 3: Ffr max = W sin  max.

But: Ffr max = sFN = s Wcos  max. Hence: W sin  max = s Wcos  max Or: s = sin  max = tan  max cos  max Thus, if the plane is gradually tipped up until the block just breaks away and the plane angle is then measured, the coefficient of static friction is equal to the tangent of this angle, which is called the limiting angle of repose. It is interesting to note that W cancelled out in the derivation of Equation 3 so that the weight of the block doesn't matter.

Exercise 1

PROCEDURE: This experiment requires you to record measurements in Newtons. Remember that in SI units the unit of force is called the Newton (N). One Newton is the force required to impart an acceleration of 1m/s2 to a mass of 1 kg. Thus 1 N = 1 kg.m/s2. You can convert any kg- mass to Newtons by multiplying the kg-weight by 9.8 m/s2, i.e., 100 g = 0.1 kg = 0.1 x 9.8 = .98 N. 1. Determining force of kinetic or sliding friction and static friction:

a. The wooden blocks provided in the LabPaq are too light to give good readings so you

need to put some weight on them, such as a full soft drink can. Weigh the plain wood block and the object used on top of the block. Record the combined weight in grams and Newtons.

b. Place the ramp board you provided horizontally on a table. If necessary tape it down

at the ends with masking tape to keep if from sliding.

c. Begin the experiment by setting the block and its weight on the board with its largest surface in contact with the surface of the board. Connect the block’s hook to the 500-g spring scale.

d. Using the spring scale, slowly pull the block lengthwise along the horizontal board.

When the block is moving with constant speed note the force indicated on the scale and record. This is the approximate kinetic or sliding frictional force. Repeat two more times.

e. While carefully watching the spring scale, start the block from rest. When the block

just starts to move note the force indicated on the scale and record. You should notice that this requires more force. This force is approximately equal to the static frictional force. Repeat two more times.

DATA TABLE 1: Mass of block: _____ Kg Weight ____ N Flat Board Force of Kinetic

Friction, N Force of Static

Friction, N Trial 1 Trial 2 Trial 3 Average

2. Determining force of kinetic or sliding friction and static friction using a different surface

area:

a. Turn the wood block on its side b. Repeat the entire process from Part 1 above three times and record the force of

kinetic and static friction for each trial. DATA TABLE 2: Mass of block: _____ Kg Weight ____ N

Flat Board – Block Sideways

Force of Kinetic Friction, N

Force of Static Friction, N

Trial 1 Trial 2 Trial 3 Average

3. Determining force of kinetic or sliding friction and static friction using different surface:

a. Determine the force of kinetic and static friction for the glass surface and

sandpapered surface blocks provided. b. To further reinforce these concepts try Parts 1, 2, and 3 experiments using the blocks

provided with at least 2 flat surfaces around your home such as carpet, rubber, tile, cork, etc. Record your findings in tables similar to Data Table 3.

DATA TABLE 3:

Surfaces Tried: Glass Surface

Force of Kinetic Friction

Force of Static Friction

Trial 1 Trial 2 Trial 3 Average

DATA TABLE 4: Surfaces Tried:

Sandpaper Force of Kinetic

Friction Force of Static

Friction Trial 1 Trial 2 Trial 3 Average DATA TABLE 5:

Surfaces Tried: Wood on Carpet

Force of Kinetic Friction

Force of Static Friction

Trial 1 Trial 2 Trial 3 Average

4. Determining coefficient of static friction using an inclined surface:

a. Place the plain block with its largest surface in contact on the board while the board

is lying flat.

b. Slowly raise one end of the board until the block just breaks away and starts to slide down. Be very careful to move the plane slowly and smoothly so as to get a precise

value of the angle with the horizontal at which the block just breaks away. This is the limiting angle of repose  max. Measure it with a protractor (see photo that follows for an alternate way of measuring the angle) and record the result. You may also want

to measure the base and the height of the triangle formed by the board, the support, and the floor or table. The height divided by the length of the base equals the coefficient of static friction.

Remember: tan θ = opposite adjacent

c. Perform two more trials. These trials should be independent. This means that in each case the plane should be returned to the horizontal, the block placed on it, and the plane carefully moved up until the limiting angle of repose is reached.

DATA TABLE 6:

Height Base Length θ max μs Trial 1 Trial 2 Trial 3 Average

Calculations: 1. Using the mass of the block and the average force of kinetic friction from Data Table 1,

calculate the coefficient of kinetic friction from Equation 1:

fr(k) = k FN. Therefore k = Ffr(k) / FN

2. Using the mass of the block and the average force of kinetic friction from Data Table 2, calculate the coefficient of kinetic friction for the wood block sliding on its side. Record your result and see how it compares with the value of k obtained from Data Table 1.

3. From the data in Data Table 3, 4 & 5 compute the coefficient of static friction, s for, the

glass surface on wood, the sandpapered surface on wood, and wood on carpet, etc from each of your three trials. Calculate an average value of s. Record your results in your own data sheets.

Ffr(s) = s FN Therefore s = Ffr(s) / FN

4. From the data obtained in Data Table 6 calculate s for wood on wood from each of your

three trials. s = tan θ = sin max or s = tan θ = height

cos max base 5. Calculate an average value of s. Record your result on the data sheet. Questions: A. How does the coefficient of static friction compare with the coefficient of kinetic friction

for the same surfaces and areas? B. Why is it important to reduce friction during the operation of machinery? C. How does grease or oil affect the coefficient of friction?

Simple Machine – Lever

Experiment Summary:

Students will have the opportunity to learn about the mechanical advantage of machines and how to calculate the efficiency of a machine. They will

construct first, second, and third class levers using both a ruler and a meter stick, test different loads in the levers, use a spring scale to measure force,

and calculate the mechanical advantage of each type of lever.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To explore the concept of mechanical advantage using levers

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 5 Quarters 1 Meter stick or yardstick 1 Paper cup 1 String 1 Pencil 1 Tape From LabPaq 1 Ruler, Metric 1 Scale-Spring-500-g

Discussion and Review

In a mechanical sense a machine is any device used to change the magnitude or direction of a force. A lever is a simple machine that magnifies force. Levers are comprised of a rigid bar (lever arm), a pivot point (fulcrum), a load force or resistance force, and an effort force. The effort force creates a torque around the fulcrum. The magnitude of this torque is dependant on the magnitude of the force and its distance from the fulcrum. This torque must be balanced by the torque created by the load force. Changing the distance from the fulcrum to the load force changes the amount of force magnification. Here are important terms regarding levers:

Fulcrum = pivot point Load Force or Resistance = the weight being lifted Effort Force = the force with which the operator pulls or pushes Load Distance = the distance the load moves (from the original position) Effort Distance = the distance the effort side of the lever moves (from the original position) Workin = Effort Force x Effort Distance Workout = Load or Resistance Force x Load Distance Din = effort distance Dout = load distance AMA = actual mechanical advantage = load force/effort force IMA = ideal or theoretical mechanical advantage = effort distance/load distance Efficiency = Workout/Workin

Neglecting any frictional losses and thinking in terms of conservation of energy we can say that:

Workin = Workout,

or since Work = Force x distance (W=Fd), one may state: (Fd)in = (Fd)out

By applying a small force through a large distance a large force is exerted through a small distance.

There are three main types of levers: first-class, second-class, and third-class. A first-class lever has the fulcrum located between the effort force and the load force on the lever arm. An example of this type of lever is a pair of pliers. A second-class lever has the load force located between the effort force and the fulcrum on the lever arm. An example of a second- class lever is a nutcracker. A third-class lever has the effort force located between the fulcrum and the load force on the lever arm. An example of this type of lever is the human lower arm; the bicep, attached just below the elbow, can be used to lift a weight being held in the hand at the end of the lower arm

In this experiment you will calculate the mechanical advantage and efficiency of a lever. You will also calculate moments (defined later). Mechanical advantage (MA) is a ratio that shows how much the machine is helping you.

MA = resistance force (load)

Effort force

As an example: Assuming there is a 2 meter long 1st class lever with the fulcrum at 0.5 m and a 12 Newton load weight on one end, how much effort force needs to be applied?

F load * d load = F effort * d effort 12N * 0.5 m = F effort * 1.5 m F effort = 6 N.m/ 1.5m F effort = 4 N MA = 12/4 = 3

A moment is a force acting at a distance that is producing a torque or twisting effect. By convention moments in the clockwise direction are positive and moments in the counterclockwise direction are negative. If a lever is not moving then the sum of all the moments must equal zero. Moments are measured in units of Newton-meters. Since work is force (in Newtons) applied over a distance (meters), the units are N.m. In SI units, the unit of work is called a Joule (J), and 1 J = 1 N.m.

PROCEDURE: For these next experiments it will be helpful to have an assistant. It is possible, but may be difficult for one person to set up these arrangements alone. Experiment 1:

1. Securely tape a round

pencil to a table so that it cannot slip. Place a ruler across the pencil so that it is balanced. If a hexagonal pencil is used, it should be taped so that the ruler rests on a corner, not a flat face. The pencil is the fulcrum of this lever.

2. Place a quarter 4 cm away from the fulcrum. This is the load. 3. Balance the load with a quarter on the other side of the fulcrum. Place this quarter

wherever necessary to balance the lever. This is the effort or force required to balance the load. Record the distance from the fulcrum.

4. Add another quarter to the load at the 4-cm mark. Re-balance the lever using just

one quarter as a force. Move it wherever required. Record the distance from the fulcrum.

5. Add a third quarter to the load. Re-balance. Can you still balance the lever with just

one quarter? Add a fourth quarter and re-balance. Record all results in a data table similar to the one that follows

6. Calculate the Ratio of distances which equals Effort Distance/Load Distance.

DATA TABLE 1: Fulcrum at _______ cm

Trial Load (Mass)

Distance of Load from fulcrum

Effort (Mass)

Distance of Effort

from fulcrum

Ratio: Effort Distance/ Load Distance

1 1 quarter 1 quarter 2 2 quarters 1 quarter 3 3 quarters 1 quarter

4 4 quarters 1 quarter

Experiment 2: Part I - First-class lever:

1. To perform this procedure tie and tape a string around a meter stick at the fulcrum point for each setup. If a meter stick is not available, use the longest ruler you can find, but the shorter your ruler, the less sensitive and less accurate will be your results. You may also use a “yard” stick and convert the inches into metric measurements or tape your meter tape to its backside. Remember: 1 inch = 25.4 mm = 2.54 cm.

2. Suspend the meter stick by the string so that it hangs freely. Possible places to

suspend it from include a clothes rod, a shower curtain rod, and the handle on a kitchen cabinet. It is important that the meter stick assembly hangs freely and that you are able to use and read the spring scale with it. The assembly should not rest against a door face or anything else.

3. Make a load pan from a paper cup by attaching a piece of string to the rim of the

paper cup in such a way that the cup can hold weights while being suspended from the meter stick. At the free end of the string make a loop that will slip over the meter stick and allow the load pan to be moved and to swing freely.

4. Make a first class lever by suspending the load pan at the 20-centimeter mark. Tape

the string in place so the load pan does not slip. Place a known weight (150 – 200 g) into the load pan. You may also tie some string around a small household item of the appropriate weight, form a loop and hang it on the meter stick. A small spice glass jar has about the right weight. Use the 500-g spring balance to determine the mass of your “weights”. The spring scale will be accurate to +/- 2.5 g, which is adequate for this experiment.

5. Attach the 500-g spring scale at the far end of the meter stick so that the weight-

bearing hook is at the bottom. When the assembly is steady, gently pull downward on the spring scale until the meter stick is level and parallel to the floor. Read the scale to measure the force required to balance the load. The pulling force plus the mass of the spring balance represent the “effort force” Record all data values in a table. Include load, distance from fulcrum to load, effort, and distance from fulcrum to effort. The spring scale calibration is in grams and Newton. Since the gram scale is easier to read, it may be best to record the results in grams and then convert to Newtons. Remember that to convert grams to Newtons the formula is: (g/1000)* 9.8 (Force = mass in grams x gravitational constant / 1000)

6. Change the weight in the load pan, measure the force required to balance the new

load, and record the data in your table.

7. Move the load pan to the 10-centimeter mark and tape it in place. Again measure the force required to balance the load and record the data in your table.

8. Move the load pan to the 5-centimeter mark and tape it in place. Again measure the

force required to balance the load and record the data in your table.

DATA TABLE 2: First-class Lever, Fulcrum at _____ m Trial Load

(Mass, g)

Load (Mass,

Load distance,

Mass of 500-g

Spring scale

reading, N

Effort Force, N

Effort Distance,

M.A.

1 62g = 0.61N

2 62g = 0.61N

3 62g = 0.61N

Example Data Table

Trial Load (Mass,

Load (Mass,

Load distance,

Mass of 500-g

Spring scale

reading, N

Effort Force, N

Effort Distance,

M.A.

1 100 1 0.3 62g = 0.61N 10g =0.1N 0.71N .45

1.41

2 153 1.5 0.3 62g = 0.61N 45g =0.44N 1.05N .45

1.42

Checking results: Workin = Workout or 1N*0.3m = 0.71N*.45m

* MA = 1/0.71 = 1.41

Experiment 2: Part II - Second-class lever:

1. This time attach the string to one end of the meter stick. Again you need to suspend the lever assembly where it can hang freely and the scale can be read.

2. Tape the load pan in place 20 cm from the fulcrum string and place a known force of

weights (150 – 200 g) in the load pan.

3. Attach the spring scale (effort) at the opposite end of the meter stick, but the scale must now be above the meter stick to pull upward and balance the load.

4. Gently pull upward on the meter stick until it is level and parallel to the floor. Read

the scale and record the effort.

5. Make several additional trials with the load pan positioned at different distances

from the fulcrum, such as 10 cm, 20 cm, 30 cm, 40 cm, etc. Record the data values, including load (resistance), distance from string to load, effort, and distance from string to effort.

DATA TABLE 3: Second-class Lever, Fulcrum at _____ m

Trial Load (Mass, N)

Load distance, m

Effort Force, N

Effort Distance, m

M.A.

Example 1.47 0.2 80g = 0.78N .90 1.9 1 2 3 4

etc

Experiment 2: Part III - Third-class lever:

1. To make a third-class lever, securely tape a string in the middle and at the end of the meter stick. Tie the string from the meter stick to a door knob or similar support as shown below; the meter stick will hang vertically when not supported. Tape the load pan to the opposite end of the meter stick; then use the scale to pull the meter stick to a level position; the scale will be close to the fulcrum and between the fulcrum and the load. The load (resistance) will be out on the end of the lever arm.

2. Read the scale and record the effort from three different distances between the

fulcrum and the load. You will see that this lever will be less efficient than the other two levers, but it is still an important configuration. Your elbow works quite well on this principle, as does a crane used in heavy construction.

3. Record all data values, including load, distance from fulcrum (string) to load, effort,

and distance from string to effort.

Tie string to door knob or similar Load or

Resistance

Scale

Third-class Lever

Effort

DATA TABLE 4 (Third-class Lever), Fulcrum at _____ m Trial Load

(Mass, N) Load distance,

M Effort

Force, N Effort

Distance, m

M.A. 1 2 3

Calculations: 1. In Experiment 1 calculate the ratios of the measured distances; i.e. the rations of Effort

Distance/Load Distance 2. In Experiment 2, Parts I, II, & III convert grams as needed to Newtons.

3. In Parts I, II, & III calculate M.A. for each trial of each lever type. Questions:

A. In Experiment 1 you calculated the ratios of the measured distances, i.e. the ratios of Effort Distance/Load Distance. What is the significance of these ratios? How did your calculations compare to your expectations?

B. The spring balance is reasonably accurate for determining the load mass. However,

the spring balance weighs 62 grams. Explain how to use the Workin = Workout principle to verify the mass of the spring balance.

C. After examining the 1st class lever data what kind of general statement can be made

with regards to mechanical advantage and the relationship of load distance to effort distance?

D. What happens to the mechanical advantage for 2nd class levers as the load moves

further away from the fulcrum?

E. What is the significance of the mechanical advantage of class 3 levers?

F. What class lever is represented by a fishing pole? Why?

G. What kind of lever is represented by an oar used in rowing? Why?

Simple Machine – Pulleys

Experiment Summary:

Students will have the opportunity to construct a simple machine using pulleys rigged with strings. They will calculate the mechanical advantage with

varying amounts of strings. They will examine a single fixed pulley, a single movable pulley, and double pulleys and calculate the work, efficiency, and

mechanical advantage of each system.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To investigate the properties of a pulley and its use as a simple machine.

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 1 Support for pulleys 1 Small plastic bag 1 Tape 1 Coins or washers for weights From LabPaq 2 Pulleys 1 Scale-Spring-500-g 1 Tape measure, 3-m Dissection Tray Misc Supplies PK-W 1 String - Qty-Length in Meters

Discussion and Review

A pulley is a type of simple machine consisting of a grooved wheel turning on an axle. It will reverse the direction and reduce the amount of the force required to lift or move an object. Pulleys can be used singly or in combination to do work. A pulley system is simply one or more pulleys connected by rope or string. If energy is conserved for a machine then the work done by the machine must be equal to the work put into the machine: Work out = Work in. The work done by a pulley equals the weight it lifts (W = mg) times the height it lifts it (h). The work that you put into the machine equals the Force (F) that you exert on the string times the distance that you pull the string (d). So, for an ideal pulley:

Fd = Wh (= mgh) Of course there is some friction present in any real pulley so we would expect that some of the work that we put into the machine would be dissipated by friction, mostly as heat energy. So for a real pulley, Fd = Wh + Work done against friction. The theoretical mechanical advantage (T.M.A.) of a machine is the ratio of the ideal output force to the ideal input force, or: T.M.A. = Fout theoretical = d in Fin theoretical d out The theoretical mechanical advantage of pulley systems is easy to determine: Count the number of rope or string segments on each side of the pulleys, including the free end. If the free end is to be pulled down, subtract 1 from this number. This number is the mechanical advantage of the system! To compute the amount of force necessary to hold the weight in equilibrium, divide the weight by the mechanical advantage. The theoretical mechanical advantage is also called the ideal mechanical advantage (I.M.A.). In real situations friction al losses occur and Actual Mechanical Advantage (AMA) is defined as: A.M.A = F out (resistance) F in (effort) The ratio of useful work done by the pulley (Wh) to the work you put in (Fd) is the efficiency of the pulley, which is usually expressed as a percent: Efficiency = Work out x 100% = Fout Dout x 100% = AMA Work in Fin Din TMA

PROCEDURE: Before beginning, you must first find a suitable horizontal support from which to freely hang your pulleys. Ideally there should be a wall close behind the support so you can affix paper or a meter tape for marking distances. A bathroom or kitchen towel bar is ideal for this purpose. It is important that the pulley assembly hangs freely, that you are able to read and record measurements from the spring scale, and that the assembly does not rest against anything. You must also make a weight bag to use as the mass. Place coins and/or washers that make a force of less than 4.9 N (500 g) inside a small plastic bag. Record the bag’s weight in the data table as resistance. Tie a piece of string around the top of the bag so the weights cannot fall out and suspend the bag as shown in the following experiment diagrams. 1. Single Fixed Pulley: (I.M.A.=1)

a. Suspend a single fixed pulley from a support

by tying a string from the support to the pulley as shown.

b. Pass a second single string through the pulley. At the bottom end of

the string attach the weight bag and at the other end of the string attach the spring scale.

c. Start with the weight bag suspended off the ground. Note the

exact location of the bag and the scale. Pull on the scale until the weight bag rises 10 cm from its starting point. Record this as the resistance distance.

d. Read the scale and record this as the effort. Also record the effort distance which is

the distance which the scale moved.

2. Single Moveable Pulley: (I.M.A = 2)

a. Pass a single string through the pulley as shown in Figure 2. Attach one end of the string to the support and the other end to the spring scale. Attach the weight bag directly to the bottom of the pulley. Note the exact location of both the bag and the scale, and record the scale reading.

b. Lift the weight bag by lifting the scale. Measure the

distance that the resistance moves and record this as the resistance distance. Measure the distance that the scale moved and record this distance as the effort distance.

3. Double Pulleys: (I.M.A. = 2)

a. Tie one pulley directly to the support. Tie the end of a

string to the end of that pulley. Run the other end of the string around a second pulley and then up and around the first pulley. Tie this end to the scale. Attach the weight bag to the lower pulley.

b. Lift the weight bag off the ground some distance and

note the exact location of the bag and the scale. Pull the scale to lift the weight further; measure and record that resistance distance. Read the scale and record this effort force. Measure and record the distance that the scale moves, the effort distance.

DATA TABLE:

1 Fixed Pulley 1 Movable Pulley 2 Pulleys Effort (g or N) Effort distance (m) Resistance (g or N) Resistance distance (m) Work in Work out Efficiency Mechanical advantage

Calculations: 1. Calculate Work out and Work in:

Work out = Weight lifted x height lifted (N.m) Work in = Force on string (from spring scale) x distance pulled (N.m)

2. Calculate the actual mechanical advantage of the three pulley arrangements:

A.M.A. = resistance/effort

3. Calculate the efficiency of the three-pulley arrangements. Efficiency = AMA x 100% IMA

Pendulum and the Calculation of g

Experiment Summary:

Students will have the opportunity to learn about pendulums, including how to calculate the period of a pendulum in relation to its length. They will

construct pendulums of varying lengths and vary the position from which they are pulled. Students will then calculate the force of gravity.

Peter Jeschofnig, Ph.D. Version 09.1.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To calculate the acceleration due to gravity by observing the motion of a pendulum.

Materials

Materials From: Label or Box/Bag: Qty Item Description:

Student Provides 1 Support for the pendulum 1 Weights, coins, or washers 1 Small plastic bags 1 Tape From LabPaq 1 Protractor 1 Scale-Spring-500-g 1 Stopwatch-digital 1 Tape measure, 1.5-m 1 Tape measure, 3-m String & Weight Bag String & Weight Bag 1 String - Qty-4.0 Meters

Discussion and Review

A pendulum is a weight hanging from or supported at a fixed point so that it swings freely under the combined forces of gravity and momentum. A typical simple pendulum consists of a heavy pendulum bob (mass = ) suspended from a light string. It is generally assumed that the mass of the string is negligible. If the bob is pulled away from the vertical with some angle, , and released so that the pendulum swings within a vertical plane the period of the pendulum is given as: Equation 1:

where “L” is the length of the pendulum and “g” is the acceleration due to earth's gravity. Note that only the first three terms in the infinite series are given in Equation 1. The period is defined as the time required for the pendulum to complete one oscillation. That is, if the pendulum is released at some point, P the period is defined as the time required for the pendulum to swing along its path and return to point, P. The above formula for the period of the pendulum is greatly simplified if we limit the initial angle θ to small values. If θ is small we can approximate the period of the pendulum with a “first-order expression”, which in the case of our simple pendulum is given as:

Equation 2:

Note that the period in this expression is independent of the pendulum's mass at initial angle, θ. Also, it is important to understand that the above equation is valid only for small angles and is substantially less accurate with large angles. During the cyclic swinging motion of a pendulum there is a constant yet gradual change of kinetic energy to potential energy and back to potential kinetic energy. In order to describe this phenomenon here are some terms you need to know:

Amplitude: The distance the pendulum travels from the center point out to the point of maximum displacement. Frequency: The number of complete cycles per unit of time. Periodic motion: The type of motion in which the object returns to the point of origin repeatedly. Because of the rotation of the earth a pendulum will be slightly deflected on its course on every circuit. This is observable on a very long pendulum called a Faucault pendulum. Look up Faucault pendulum. Period (T): The length of time for one trip, back and forth. Displacement: The distance from the center point, straight down. Cycle: One swing of the pendulum back and forth. Bob: The mass on the end of the pendulum.

PROCEDURE: Before beginning, you must first find a suitable support from which to freely hang your pendulum. Ideally there should be a wall close behind the support so you can easily affix your protractor and meter tape for recording movements. A bathroom or kitchen towel bar is ideal for this purpose. Or you might rig a support like the one at right and place it on a narrow shelf or table top. The important things are that your support allows the pendulum to hang freely; that you are able to read and record measurements from the protractor and meter tape; and that pendulum string not touch anything or be obstructed from any direction.

You will also need to make a weight bag to use as the bob. Place coins, weights, or washers totaling around 25 grams inside a small plastic bag. Tie a short piece of string around the top of the bag so the weights cannot fall out Weigh your bob and record the weight. Note: one quarter should weigh around 5.7 grams. 1. Weigh your bob and record its mass. 2. Suspend the bob from a string that measures exactly one meter (100 cm) between

where it attaches to the support and where it attaches to the center of the weight bag you are using as a bob. To accomplish this, you obviously must start with string that is longer than a meter.

3. Securely affix a protractor behind where the string is attached to its support so you will

be able to measure the pendulum’s amplitude in degrees. 4. Stretch a meter tape horizontally and securely affix it so that its 50-cm mark is directly

behind the bob at rest. 5. Observe the protractor and pull the bob out to the 5o-mark. Then observe the meter tape

and record the distance in cm of the bob displacement. 6. With a stopwatch in your other hand, release the bob and time how long it takes for the

bob to move through 5 complete cycles. Record the time in Table 1. Perform two more trials from the 5o-mark. Record each time, then average the three trials and calculate the period for one cycle.

7. Repeat this procedure and record results for each of the angles shown in Table 1.

DATA TABLE 1: Length of string: _____ cm = _____ m Mass of bob: _____ g = _____kg

Amplitude Amp. Trial 1 - seconds

Trial 2 - seconds

Trial 3 - seconds

Avg. Time Period

Degrees cm 5 cycles 5 cycles 5 cycles 5 cycles 1 cycle 5 o 10 o 15 o 20 o 25 o 30 o

8. Place double the bob weight into a second plastic bag and repeat this procedure using a

10o.amplitude Record the data in Table 2. DATA TABLE 2: Length of string: ________ cm = _______ m Amplitude: _______o

Bob Weight Trial 1 Trial 2 Trial 3 Avg. Time Period Grams

9. Put the original bob back on your pendulum. Use a 5o or 10o amplitude and make three

trials each with successively shorter lengths of string, i.e., 100 cm, 75 cm, 50 cm and 25 cm. Record this data in Table 3.

DATA TABLE 3: Mass of bob: ________ g = _______ kg Amplitude: _______o

Length (m) Trial 1 Trial 2 Trial 3 Avg. Time Period .25 .50 .75 1.0

Calculations: Solve the pendulum formula for g. Substitute the data you recorded for the values for t and L (length of string) in the formula. Calculate to the correct significant figures. Then calculate your percentage error as compared to the accepted value for g. The accepted value of g is 9.8 m/s2.

t = 2 π √(length/g)

g = (2π)2 L t2

where: g = acceleration due to gravity t = time in seconds L = length of pendulum string in meters

Note: If you get very large errors in this lab you are doing something wrong. Your calculations need to be double-checked. Questions:

A. How did the change in the weight of the bob affect the resulting period and frequency?

B. How did the change in amplitude affect the resulting period and frequency?

C. How did the change in length of the pendulum affect the period and frequency?

D. What would happen if you used very large amplitudes? Check your hypothesis by

trial. What amplitude did you use? What is the result?

E. Hypothesize about how a magnet placed directly under the center point would affect an iron bob? Try it and find out. Did your trial verify your hypothesis?

F. How close was your calculation of the value of g at your location? What might be a

few sources for error in your experimental data and calculations?

G. What would you expect of a pendulum at a high altitude, for example on a high mountain top? What would your pendulum do under weightless conditions?

©Hands-On Labs, Inc. LabPaq PK-1 145

Centripetal Acceleration

Experiment Summary:

Students will have the opportunity to assemble a centripetal apparatus to study the forces involved in rotational motion. Students will use varying

masses to measure the time it takes for one revolution and use the data to derive the radius, theoretical centripetal force, and velocity. They will explore and graph the relationships among velocity, centripetal force, and the radius

of the circular path of an object undergoing constant circular motion.

Peter Jeschofnig, Ph.D. Version 09.1.02

Review the safety materials and wear goggles when working with the centripetal

acceleration apparatus. Read the entire exercise before beginning. Take time to organize the materials needed and set aside a safe workspace in which to complete the exercise.

Objectives

©Hands-On Labs, Inc. LabPaq PK-1 146

 To calculate the angular velocity of a spinning object using varying hanging and rotational masses and varying radii.

 To calculate the theoretical centripetal force.  To calculate the experimental centripetal force.  To graph and analyze experimental data.

Estimated time required to complete this Experiment: 3 hours

Materials

©Hands-On Labs, Inc. LabPaq PK-1 147

Materials from: Qty Item Description

Student 1 Assistant

LabPaq 1 Centripetal force apparatus - including: Glass Rod, Steel Washers (20), Paper Clips (2), Rubber Stopper- 2 Hole, Thread

1 Scale-Spring-500-g 1 Stopwatch-digital 1 Tape measure, 3-m

Discussion and Review

©Hands-On Labs, Inc. LabPaq PK-1 148

Newton's first law of motion states that each body moves at a constant speed in a straight line unless a net external force acts upon the body. Thus if an object is moving around a circle, even at constant speed, there must be a net external force acting on the object to accelerate the object (in this case, changing its direction). The acceleration on a rotating object points toward the center of the circular motion and is called centripetal acceleration. The term centripetal comes from the Latin words centrum and petere, which mean "center- seeking."

Figure 1 – Centripetal force (Wikimedia Commons)

A centripetal force is a force that is always directed toward the center of the circular path. A body of mass m, moving in a circular path of radius r, with a speed v, has a centripetal acceleration a = v2/r. From Newton's second law of motion the force required to give the mass this acceleration is F = ma = mv2/r.

Item Description

1 Axis

2 Centripetal force or acceleration

3 Velocity

ω Angular velocity

Some amusement park rides use centripetal acceleration to increase the thrill of the ride. In one type of ride, people stand against the wall while the ride spins. After reaching a certain velocity, the floor drops away and the people

stay in place because they are pushed against the wall. If the wall did not hold them in place, they would fall off at tangents from the ride. See if you can match the forces and accelerations of the ride to the diagram in Figure 1.

©Hands-On Labs, Inc. LabPaq PK-1 149

There are two different ways to measure a rotational angle (abbreviated as θ): degrees and radians. A revolution (rev) is defined to be one complete turn, and one complete turn is defined to be 360 degrees (°), so 1 rev = 360°. A radian measure is defined in terms of the ratio of two important lengths: the radius r, and the arc length s. Thus, θ in radians = s/r. See Figure 2. One radian (rad) is the angle subtended from the center of a circle by an arc whose length is equal to the radius of the circle. Thus 2 rad is equal to 360°and 1 rad = 360/2 ≈ 57.3 o. See Figure 3.

Figure 3 – Conversion chart between degrees and radians.

©Hands-On Labs, Inc. LabPaq PK-1 150

There are many important examples of centripetal acceleration. For example, Earth and the Moon exert gravitational forces on each other, and the Moon experiences a centripetal acceleration toward the center of the Earth-Moon system. In this experiment, a rubber stopper is connected to a string and is set in rotation by spinning it manually in a horizontal circle. The tension in the string causes the stopper to undergo centripetal acceleration. (If the string suddenly breaks, the rubber stopper will move in a straight line in agreement with Newton’s first law.) If an object of mass m is rotating with constant speed v about a circle of radius r, then the centripetal acceleration on the object has the magnitude:

a = v2/r = (rω)2/r = ω2r where:

 ω is the angular velocity in radians per second (rad/s).  ω = θ/T  θ = angular displacement (rad),  T = time (s)

For the purpose of this experiment, the time it takes to complete one revolution will be called T. The speed of the rotating object is calculated by dividing the angular displacement, in this case the circumference of the orbit C, by the period T.

ω = C/T = 2r/T

Therefore, to measure the time T required to make one revolution and to find the angular velocity ω use the following equations:

 The circumference of the circle = distance of 1 rev = 2r  Period, T = time required for 1 complete revolution, and  The velocity of the rotational object = distance traveled/time = 2r/T.  Thus, the centripetal force that must be supplied is given by F = mv2/r.  Centripetal acceleration = a = (2r/T)2r = 42r/T2

Centrifugal force is a hypothetical outward force which an object traveling on a curved path experiences. What actually takes place is the action of centripetal acceleration pulling toward the center of the curved path, but what the object apparently experiences is a force pushing outward from the center of the motion. Because the result is an outward force, many uses for this action have been devised. One example of a device that uses the centrifugal force is a centrifuge.

Centrifuges are commonly used in laboratories to separate mixtures based on density. Solutions are placed in centrifuge tubes and then into the centrifuge. These solutions spin at high speeds varying from 400 to 100,000 revolutions

per minute (rpm). The more dense substances fall to the bottom of the centrifuge tubes.

©Hands-On Labs, Inc. LabPaq PK-1 151

In a classic physics experiment demonstrating centrifugal force, students whirl a bucket or cup of water overhead so that the bucket is turned upside down. While the bucket is upside- down, none of the water will spill out. See Figure 4.

Figure 4 – Demonstrating centripetal acceleration while spinning water in a cup

Exercise 1

©Hands-On Labs, Inc. LabPaq PK-1 152

In this experiment, you will study how varying the length of the radius of a rotating mass will affect the period. PROCEDURE: WARNING: Choose an area that is free from obstructions and breakable objects, such as a field outside. You will be swinging weights on a string and if these weights were to break free, they could potentially hit objects or people. Choose an area where only your assistant is present to reduce the risk of people being injured. Wear goggles so that the rotating stopper does not hit your eyes. 1. Prepare a table similar to Data Table 1 below to record observations while performing

the experiment.

Data Table 1 – Mass of washers and rubber stopper Item Value

Mass of rubber stopper (kg) Mass of all washers combined

(kg)

Number of washers Average mass of each washer

(kg)

2. Use the spring scale to determine the mass of the rubber stopper and the washers in

your centripetal apparatus kit. Record the mass of the stopper in Data Table 1.

3. The scale is not sufficiently accurate to measure individual washers, so you should measure all washers at one time. Record the number of washers in Data Table 1. Place all of the washers into a bag to weigh their mass, and record the total mass in Data Table 1. Find the average mass of each washer, and record in Data Table 1. The mass of the washers vary from 3.79 grams (g) (3.79 ×10-3 kg) to 4.1 g (4.1 x 10-3 kg) each. Convert the units to kilograms and record the mass into Data Table 1.

4. Prepare a data table similar to Data Table 2 shown below. Data Table 2: Varying radius data

Constant hanging mass (approximately 30 g): _______ (kg) Constant rotating mass: Use mass of stopper from Data Table 1: ___________ (kg) A B C =

B/10 D =2×A E = D/C F = E2

Radius (m)

Time (s) 10 rev

Time (s) 1 rev

Circumference 2r (m)

Velocity (m/s)

(Velocity)2 (m2/s2)

Trial 1 Trial 2 Trial 3 Trial 4

©Hands-On Labs, Inc. LabPaq PK-1 153

5. Assemble the centripetal apparatus as shown in Figure 5.

a. Use the measuring tape to measure 1.2 m of the string and cut it. This

portion of the string will be used for this experiment.

b. Before threading the string through the glass rod, make sure that the smoother end of the glass rod is at the top nearest the rotating rubber stopper.

c. Tie the rubber stopper to the string and thread the other end of the string through the plastic-covered glass rod.

d. Thread about 30 g of washers (approximately 8 washers) onto the end of the

string opposite from the stopper. Record this constant hanging mass in the line above Data Table 2.

e. Tie the washer-end of the string to a paper clip, and if needed, spread open

the paper clip to ensure that the washers do not fly off the apparatus. See Figure 6.

f. Tie the paper clip about 20 cm above the washers. When finished, your

apparatus should look like the one in Figure 7. Figure 5 – Centripetal force apparatus

Number Explanation

1 Rubber stopper

2 Glass tube covered with plastic

3 String 4 Paper clip

5 Washers

L Length of string

©Hands-On Labs, Inc. LabPaq PK-1 154

Figure 7– Assembled centripetal apparatus

6. Pull the string through the glass rod so that

approximately 0.8 m of string is between the glass rod and stopper. Practice swinging the stopper around in a circle while holding onto the glass rod. Support the suspended mass containing the washers with one hand and hold the rod in the other. Be careful! Review safety precautions mentioned at the beginning of this procedure. See Figure 8.

7. Swing the stopper in a circular motion. Slowly

release the hanging mass and adjust the rotating speed of the stopper so that the paper clip attached to the string above the washers stays a few centimeters below the bottom of the tube, neither rising nor falling. Do not move your hand too much while swinging the stopper. Ideally, the steel washers should be stationary. Keeping your hand steady will help the rubber stopper move smoothly. Practice stopping the spin while simultaneously grasping the string just above the tube. This action will allow you to measure the radius of the spin circle, which is the length of the string from the top of the tube to the center of the stopper.

8. Stop spinning the rubber stopper and use the measuring tape to measure the length of

the string in meters, shown as L in Figure 5. Record this length as the radius for Trial 1 in Data Table 2.

©Hands-On Labs, Inc. LabPaq PK-1 155

9. After you are able to spin the stopper with a steady pace, you can begin the experimental

portion of the lab.

10. Begin to spin the apparatus, maintaining a constant radius. After the spin is stabilized have an assistant use a stopwatch to time (in seconds), 10 revolutions. Record this 10-rev time for Trial 1 in Data Table 2.

11. Shorten the length of string between the stopper and the top of the glass tube by 5 to 10 cm. Pull the string through the bottom of the glass tube to shorten the distance L between the top of the glass tube and the stopper. Use the tape measure to record this new length between the top of the glass rod and the stopper as the radius for Trial 2 in Data Table 2.

12. Repeat the procedure of swinging the stopper for 10 rev while it is being timed. Record

this time for 10 rev in Data Table 2 for Trial 2.

13. Shorten the string by another 5 to 10 cm as in Step 10, and record this new radius in Data Table 2 for Trial 3.

14. Repeat the procedure of swinging the stopper for 10 rev while it is being timed. Record

this time in Data Table 2 for Trial 3. 15. Shorten the string by yet another 5 to 10 cm as in Step 10. Record this new radius in

Data Table 2 for Trial 4.

16. Repeat the procedure of swinging the stopper for 10 rev while it is being timed. Record this time in Data Table 2 for Trial 4.

17. You will complete the calculations for columns C through F later, in Exercise 3.

Exercise 2

©Hands-On Labs, Inc. LabPaq PK-1 156

In this experiment, you will study how mass affects centripetal force. The radius will be kept constant at 0.5 m while the hanging mass will increase. PROCEDURE:

1. Prepare a data table similar to Data Table 3 shown below.

Data Table 3 – Varying hanging mass data

Constant rotating mass: The mass of stopper from Data Table 1 ___________ (kg).

A B C = B/10

D =2×A E = D/C F = E2

Hanging Mass (kg)

Radius (m)

Time (s) 10 rev

Time (s) 1 rev

Circumference 2r (m)

Velocity (m/s)

(Velocity)2 (m2/s2)

Trial 1 Trial 2 Trial 3 Trial 4

2. Adjust the radius of the rotating mass to 0.5 m. Because this value will remain the same for this part of the experiment, record the length of the radius in Data Table 3 for Trials 1 through 4.

3. Change the number of hanging washers so that they weigh approximately 40 g. Record this hanging mass in Data Table 3 for Trials 1 and 2. Record the mass of the stopper in Data Table 3. Use the mass for the stopper from Data Table 1.

4. Use this 40-g hanging mass to perform two trials of 10 rev each in a manner similar to that in Exercise 1. Record the time for 10 rev in Data Table 3 for Trials 1 and 2.

5. Add more washers until the hanging mass is approximately 50 g. Record this hanging mass in Data Table 3 for Trials 3 and 4.

6. Use this 50-g hanging mass to perform two trials of 10 revolutions each in a manner similar to that in Part 1. Record the time in Data Table 3 for Trials 3 and 4.

7. You will complete the calculations for columns C through F after Exercise 3.

Exercise 3

In this experiment, you will study how mass affects centripetal force. You will keep the radius constant while increasing the rotating mass. PROCEDURE: 1. Prepare a data table similar to Data Table 4 shown below. Data Table 4: Varying rotating mass data

Constant hanging mass: _______kg

A B C = B/10 D = 2×A E=D/C Rotating

Mass (kg)

Radius (m)

Time 10 rev (s)

Time 1 rev (s)

Circumference 2r (m)

Velocity (m/s)

Trial 1 Trial 2 Trial 3 Trial 4 2. Keep the hanging mass at its current weight of 50 g and record this in the blank above

Data Table 4. Use the radius at its current length of 0.5 m, and record this value into Data Table 4 for Trials 1 through 4.

3. Untie the string attached to the stopper. Tie approximately 5 g of washers together with

the stopper. Use the scale to weigh the stopper and washers together. Record this total rotating mass in Data Table 4 for Trial 1.

4. Use this rotating mass to perform one trial of 10 rev in a manner similar to the process in

Part 1. Record the time for 10 rev in Data Table 4 for Trial 1. 5. Add an additional 5 to 10 g of washers to the rotating mass. Reweigh the stopper and

washers to find this new mass. Record this rotating mass in Data Table 4 for Trial 2.

6. Use this rotating mass to perform one trial of 10 revolutions in a manner similar to that in Part 1. Record the time for 10 rev in Data Table 4 for Trial 2.

7. Repeat Steps 4 and 5 two more times, adding more weight, and recording data in Data

Table 4 for Trials 3 and 4.

Calculations: Complete the calculations for Data Tables 2, 3, and 4 as follows: 1. Calculate the average time for 1 revolution by dividing the total time by the number of

revolutions. The number of revolutions timed is 10, so divide by 10. Enter this value in column C for Data Tables 2, 3, and 4.

2. Calculate the circumference of the path of the rotating mass using the formula 2πr

Use the radius values from Column A to calculate this value and record in column D.

3. Calculate the velocity by taking the circumference and divide by the time for 1 rev

(circumference/T). Record this value in column E.

4. For Data Tables 2 and 3, square this velocity and record in column F. 5. Prepare data tables similar to Data Tables 5, 6, and 7 shown below. Data Table 5 – Varying radius data (Exercise 1)

Data Table 6 – Varying hanging mass data (Exercise 2)

Data Table 7 – Varying rotating mass data (Exercise 3)

Theoretical Fc (N)

Velocity2 radius

Experimental Fc (N)

Trial 1 Trial 2 Trial 3 Trial 4

Theoretical Fc (N)

Velocity2 radius

Experimental Fc (N)

Trial 1 Trial 2 Trial 3 Trial 4

Theoretical Fc (N)

Velocity2 radius

Experimental Fc (N)

Trial 1 Trial 2 Trial 3 Trial 4

6. Calculate the theoretical centripetal force (Fc) = hanging mass × g (acceleration due to gravity). For each trial in all three exercises, calculate this theoretical centripetal force by multiplying the hanging mass by 9.81 m/s2. Record these values into Data Tables 5, 6, and 7.

7. Calculate the experimental centripetal force for each trial in all three exercises.

Calculate the experimental centripetal force by multiplying the rotating mass by the velocity squared divided by the radius. See the equation below:

rotating mass velocity radius

F  2

Record these values into Data Tables 5, 6, and 7.

Graphs: Preparing the following graphs will help you better understand the relationships among centripetal force, velocity, and the radius of the circular path of an object undergoing constant circular motion. Normally we plot the independent variable on the x-axis and the dependent variable on the y-axis, but in this lab we will reverse this convention. You should plot the dependent variable (velocity) on the x-axis so that it is easier to see the relationship between the variables.

Plot the following graphs:

From Data Table 2:  Radius vs. velocity  Radius vs. velocity squared

From Data Tables 3 and 6:

 Centripetal force vs. velocity  Centripetal force vs. velocity squared

From Data Table 4:  Rotating mass vs. velocity  Rotating mass vs. 1/velocity squared

Questions:

A. What is the relationship between the radius and the velocity of a rotating object?

B. What is the relationship between the velocity of a rotating object and the centripetal force exerted on it?

C. The Moon orbits Earth at a distance of about 3.84 × 108 m in a path that takes 27.3

days to complete. What is the centripetal acceleration of the Moon?

D. What is the relationship between the mass of a rotating object and its velocity?

E. When graphing the values of velocity versus velocity squared, did you notice any differences? Explain your answer.

When a bucket of water is swung upside-down in a vertical circle, the water will remain in the bucket if the velocity is kept high enough. If you let the bucket slow down too much, the water will spill out. The critical velocity is the slowest velocity necessary to keep the water in the bucket. What is the critical velocity if the formula is: velocity rg , where the radius of your arm swing including the bucket is r, and g is the acceleration of gravity? Hint: You must measure or estimate the length from your shoulder to the center of the bucket for the radius.

Hooke’s Law

Experiment Summary

Students will investigate Hooke’s law and determine the spring constant for two springs and a rubber band. Students will stretch two different springs and

a rubber band, while measuring both the distance elongated and the force required to extend the springs. From this data, students will calculate the

elastic potential energy in joules and the spring constant. Students will then compare the spring data to the rubber band.

Peter Jeschofnig, Ph.D. Version 09.2.01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe workspace in which to complete the exercise.

Objectives

 To investigate Hooke’s law and to determine the spring constant for two springs and a rubber band

Time Allocation: 1–2 hours

Materials Materials From: Label or

Box/Bag: Qty Item Description:

Student Provides 1 Small rubber band 1 Computer and spreadsheet software From LabPaq 1 Scale-Spring-500-g 1 Tape measure, 1.5-m 1 Tape measure, 3-m 1 Springs, 2 sizes-PK

Discussion and Review Hooke’s law is named after the seventeenth century physicist Robert Hooke and relates the force pulling or pushing on a spring (or other elastic material) to the amount the spring stretches or compresses. The force exerted by a spring to restore itself to its natural length is referred to as the restoring force. When a spring is stretched, as in this experiment, the restoring force is exerted inwards; if a spring is compressed, the restoring force is exerted outwards. Mathematically, the restoring force of a spring is expressed as: F kx  where F = restoring force k = proportionality constant, called the spring constant x = distance the spring has been stretched or compressed The negative sign indicates that the restoring force acts in the direction opposite of the displacement direction. Depending on material, length, diameter, and number of coils, each spring has its unique spring constant. The greater the spring constant, the stiffer the spring (the more difficult it is to stretch it or compress it). The elastic limit is the maximum extension to which a spring can be stretched without permanent deformation and still return to its original shape. If a spring is stretched beyond its elastic limit, it will not return to its original shape and will remain deformed. On a force versus elongation graph, the elastic limit will show up as the point where the slope of the line changes or where the straight-line portion of the graph ends. Not all elastic materials obey Hooke’s Law. For example, rubber is generally considered a hyperelastic, neo-Hookean material because its elastic behavior varies with loading rate and temperature. Under simple experimental conditions, rubber bands seem to follow Hooke’s law for a limited range. Depending on the latex and rubber, a rubber band may not return to its exact original shape after stretching. Hooke’s law can be used in two ways. The first is to find the force exerted by a spring. The second is to derive the period of oscillating motion for a mass connected to a spring. The two related equations are: Equation 1: spring F k x  

Equation 2: 2 mT k



Where T is the time period of one oscillation cycle (a complete up and down movement of the weighted spring) and m is the mass on the end of the spring.

Commonly, a Hooke’s law experiment is conducted by adding increasing masses to a spring and recording the cumulative stretch (elongation) of the spring. This experiment will use a spring scale in place of calibrated weights to increase the force on a spring. However, this method will add an additional step to the experiment. To cancel out the effect of the internal spring of the spring scale, you need to measure the elongation of the spring for each force increment by recording the position of the top and bottom of the spring. If you only record the bottom position of the spring, you would measure the combined spring constant of the spring and the internal spring of the spring scale.

Figure 1. Stretch test.

Exercise 1 PROCEDURE: Ensure that you do not stretch a spring beyond its capacity to recover by first performing a stretch test to estimate the spring’s full elongation capacity:

 Hold the spring at both ends and pull it apart with only moderate force, not with so much force that you permanently distort it.

 Estimate how many centimeters (cm) you were able to stretch the spring and mentally divide that stretch by the number of experimental increments you wish to test.

 For each test step, you will add only sufficient weight or force to increase the stretch by one increment. For example, if you can stretch a spring 10 cm and need 10 measurements, each experimental increment should be 1 cm and you should add enough force at each step to stretch the spring by only 1 cm.

1. Suspend the spring scale from a wall hook, doorknob, or something similar with a flat

surface behind it to which you can tape a meter tape.

2. Perform a stretch test as described above on the first spring and then suspend it from the scale as shown at right.

3. Position and affix the meter tape along the side of the spring. The

location of the beginning of the tape is not important as you will record the top and bottom measurement for each force addition.

4. Hold the bottom hook of the spring and gently pull straight down with

sufficient force to stretch the spring 1/10 of its elongation capacity. Now measure and record the position of the top and the bottom of the spring. The difference will be the exact elongation of the spring. Also, record the force required to create that elongation.

Continue to stretch the spring and record data in steps, which add sufficient force to achieve an additional 1/10 elongation. You will record 10 sets of force and elongation data. The elongations recorded at each step are already cumulative elongations.

5. Repeat the above steps with a second spring of a different stiffness. 6. Finally repeat the above procedures using a small rubber band. Then continue adding

weight until the rubber band breaks or is on the verge of breaking or nearly stops stretching with added force.

Optional exercise: As an optional exercise, you can determine the spring constant k of the internal spring of the spring scale. Data Tables. Force (N) Top position of

spring, cm Bottom position

of spring, cm Elongation, cm

Bottom reading – top reading

Data Point 1 Data Point 2 Data Point 3 Etc. Force (N) Accumulated (cm)

Elongation (stretch) Accumulated (m)

Elongation (stretch) Elastic PE (Joules)

Data Point 1 Data Point 2 Data Point 3 Etc. Calculations and Analysis:

1. For each data row in each of your tables calculate: Elastic PE = 21 2

2. For each spring and the rubber band, plot the accumulated elongation (x-axis) versus the

applied force (y-axis) on a computer spreadsheet. 3. Find the spring constant for the springs in Newtons per meter from the slope of each

graph. (Refer to the Excel tutorial in the Introduction section). Spring constant, Fk x

 ,

where F is in Newton and x is in meters. Therefore, the units are N/m. 4. Find the “spring” constant for the rubber band from the slope of the curve using the

linear portion of the graph.

Sample graph. Rubber band.

Force vs Cumulative Elongation (rubber band)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Cumulative elongation, m

Fo rc

e, N

Series1

Questions:

A. How does the relative stiffness of a spring relate to its spring constant?

B. How does PE change relative to the stretch of the spring?

C. Indicate on your graph for the rubber band where the linear behavior stops. What does this mean?

D. Which is stronger in the region where Hooke’s law is obeyed, the spring or the rubber

band? Explain.

E. Explain what happens to the “spring constant” of the rubber band for the nonlinear part of your curve.

Specific Heat Capacity of Metals

Experiment Summary:

Students will have the opportunity to learn about specific heat capacity and how to calculate it. They will set up a calorimeter to measure the heat

changes of two different metals and then calculate the specific heat of each metal. Students will compare the calculated value with given values to

determine the experimental error.

Peter Jeschofnig, Ph.D. Version 09.1.02

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe work space in which to complete the exercise.

Objectives

 To measure the specific heat capacity of two different metals.

Materials

Materials From: Label or Box/Bag:

Qty Item Description:

Student Provides 1 Cooking pot for boiling water 1 Stovetop or heat source 1 Fork From LabPaq 4 Washers (may be separate or may be included in

the Centripetal force apparatus, depending on the LabPaq)

3 Cup, Styrofoam, 8 oz 1 Cylinder, 25 mL 1 Scale-Spring-10-g in Box Labeled Slim Pen Scale 1 Thermometer-in-cardboard-tube String & Weight Bag

String & Weight Bag

1 String - Qty-4.0 Meters

Weights Bag Weights Bag 3 Weight, 1/2 oz.

Discussion and Review

When energy in the form of heat, Q, is added to a material, the temperature of the material rises. Note that temperature, in units of degrees, Celsius (°C) or Kelvin (K), is a measure of how hot or cold a substance is, while heat, in units of joules (J) or calories (cal), is a measure of its thermal energy. 1cal = 4.19J. Specific heat capacity or simply specific heat, c, is the measure of the efficiency with which a substance can store heat energy. The greater a material's specific heat, the more energy must be added to change its temperature. As an example, the specific heat of water is 1.00 cal/g oC or 4.19 J/g oC, which means that 1.00 calorie or 4.19 Joules of heat is needed to raise one gram of water one degree Celsius. Specific heat values for various materials are listed in the table below. The heat needed to change the temperature, T, in a material of mass, m, is given by the equation

Q = mcT

where c is the specific heat capacity of the material. The change in temperature of the substance is the difference between its final and initial temperatures, T = Tf – Ti. When two substances at different temperatures come into contact with each other, heat energy is transferred between them. If you place a piece of hot metal into a container of cold water, we observe that the water and its container will become warmer while the metal will become cooler, until an equilibrium temperature is reached. According to the law of conservation of energy, the total heat energy lost by the metal is the total heat energy gained by the water and container:

- Qlost by object = Qgained by water

-mm cm Tm = mw cw Tw + mc cc Tc where:

mm = mass of the metal cm = specific heat of the metal Tm = change of temperature of the metal (Tfinal – T initial) mw = mass of water in calorimeter cw = specific heat of water, 1.00 cal/g oC Tw = change of temperature of the water (Tfinal – T initial) mc = mass of calorimeter cc = specific heat of the calorimeter Tc = change of temperature of the calorimeter (Tfinal – T initial)

This equation assumes that no heat is exchanged with the surrounding environment and that none of the materials undergo a phase change. As you remember from your text, the negative sign is used to maintain the sign convention for heat; we set the hot side to be negative because energy is leaving the hot sample. To keep the heat exchange with the environment to an absolute minimum, we are using well-insulated calorimeters for this

experiment. We will be using two Styrofoam® cups, one inside another, plus a Styrofoam® cover to minimize heat loss to the environment. Table 1: Specific heat values

Material cal/g °C J/g °C Aluminum 0.215 .900 Brass 0.092 .385 Copper 0.092 .385 Iron 0.107 .448 Lead 0.031 .130 Magnesium 0.245 1.030 Steel 0.107 .448 Zinc 0.093 .390 Styrofoam 0.27 1.131 Air 0.240 1.006 Water 1.000 4.19

Exercise 1

PROCEDURE: 1. An hour before starting the exercise, fill a glass of water with tap water and let it sit so it

can reach room temperature. Set out the metal pieces as well so they too can reach room temperature. Place a thermometer nearby so that you can record the room temperature.

2. You must first construct a calorimeter using three foam cups. One cup will be placed

inside of a second cup for better insulation; the bottom of the third cup will become the calorimeter lid through which a thermometer can pass.

3. Use scissors to cut the third foam cup so that it is only 6 cm high. Cut a small hole into

the bottom center of this cup. This shortened cup will be turned upside down to function as an insulating lid for your calorimeter. The hole will allow a thermometer to be inserted into the calorimeter so you can take periodic readings.

4. Use a graduated cylinder to exactly measure 25 mL of the room temperature water from

Step 1. Quickly pour the water into the calorimeter (doubled foam cup) and cover it with the lid constructed in Step 3.

5. Measure the temperature (oC) of the water in the calorimeter by inserting the thermometer through the hole in the lid as shown in the photo to the left. Record.

6. Use your hanging scale and thermometer to measure the temperature and determine

the mass of the first metal object you will test, the steel washers or the lead sinkers. Record. The temperature of the metals should be the same as room temperature since they have been left out at room temperature for an hour or more.

7. Use thread or string to securely tie all the metal washers together for use in the first trial

of this experiment. (Similarly use thread or string to securely string together the three small lead sinkers for the second trial.) Having these items tied together will make it easy for you to remove them simultaneously and minimize heat loss.

8. Bring a few centimeters of water to boil in a pot on a stove or heat source. Add the metal

objects to the boiling water and turn the heat down low enough to keep the water lightly boiling for 15 more minutes.

9. Determine the temperature of the boiling water and record. The maximum temperature

of the water will be the same as the maximum temperature of the metal. 10. Use a fork to safely yet rapidly transfer the string of metal objects from the boiling water

and drop it into the calorimeter containing the room temperature water. 11. Quickly cap the calorimeter with the cup lid. Put a thermometer through the hole and

extend its tip into the water of the calorimeter, but avoid touching the metal on the very bottom.

12. Observe the temperature rise, and record the maximum temperature reached. Try to

record this maximum temperature to a ¼-degree accuracy. 13. Repeat the exercise using the second metal object. DATA TABLE 1:

Object Description – First Metal Second Metal Room temperature, oC Mass of water in calorimeter, 25 ml = 25 g Mass of first metal object Starting temperature of water (room temp) Starting temperature of object (room temp) Highest final temperature of water & object

DATA TABLE 2:

Objects Mass (g) T initial T final T C (cal/g oC) Water in calorimeter

First metal

DATA TABLE 3: Objects Mass (g) T initial T final T C (cal/g oC)

Water in calorimeter

Second metal Calculations: 1. Calculate the specific heat of each metal. - Qlost by object = Qgained by water

Set up the equation: -mm cm Tm = mw cw Tw and solve for cm mm = mass of the metal cm = specific heat of the metal Tm = change of temperature of the metal (Tfinal – T initial) mw = mass of water in calorimeter cw = specific heat of water, 1.00 cal/g oC Tw = change of temperature of the water (Tfinal – T initial)

2. Since you will know which metals you have analyzed and have their specific heat values

in Table 1, you should be able to calculate the percent error of your experimentally derived value.

% error = experimental value – theoretical value x 100

theoretical value Questions:

A. Why is it a good idea to start with room temperature water in the calorimeter?

B. Why did we ignore the calorimeter in our calculation, although it is listed in the original equation?

C. When eating apple pie, you may have noticed that the filling seems to be much hotter

than the crust. Why is this? What can you conclude about the specific heat of the fillings vs. the specific heat of the crust?

D. Is the heat exchange between the metal and the water in the calorimeter by

radiation, conduction or convection? Why?

Determining the Speed of Sound

Experiment Summary:

Students will have the opportunity to learn how to measure the speed of sound in air using the resonance of longitudinal waves. Students will

determine the velocity of sound in air using a tuning fork of known frequency. They will measure the wavelength of a sound using the resonance of an air column inside a PVC pipe submerged at various depths in water. Then they

will use the data to calculate the sound wavelength and speed.

Peter Jeschofnig, Ph.D. Version 09.1.02

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set

aside a safe workspace in which to complete the exercise.

Objectives

 To measure the speed of sound in air using the resonance of longitudinal waves.

 To learn about sound and how it travels.  To learn how the speed of sound varies in different materials.  To learn about resonance and how to detect when it occurs.

Estimated time required to complete this exercise: 2 hours

Materials

Materials: Qty Item Description

Student Provides 1 1

Tall glass of water, 17-cm Partner

LabPaq Provides 1 Pipe (PVC), 10-in piece 1 Tape measure, 3-m 1 Thermometer in cardboard tube 1 Tuning Fork, 384-Hz

Discussion and Review

Sound is a mechanical wave that is transmitted through solids, liquids or gases. These waves are transmitted as a sequence of pressure changes, which move through compressible media such as liquids and gases. The temperature and pressure of a substance can affect the speed of sound as these properties affect the density of the substance. Sound cannot travel through a vacuum. Figure 1 – The top wave shows amplitude and wavelength. The bottom wave represents the compression and rarefaction of a medium that occurs as sound travels. This compression is shown by placing lines closer together and rarefaction—expansion—is shown by the placement of lines further apart.

Symbol Explanation

W Wavelength –the distance over which a wave shape repeats; units are usually in meters

A Amplitude – the height of a wave

Frequency – the number of wavelengths that occur per unit of time (such as one second); units are Hz for Hertz (1 Hz is 1 cycle/second)

C Cycle – the period of one full wavelength; in Figure 1, the distance between the two asterisks

Due to physical differences among substances, sound travels at different speeds within different substances. Generally, the denser the material, the faster sound will travel through it. Solids have a high density, so sound travels faster in solids than in liquids or gases. See Table 1 for examples of these different speeds. Table 1 Speed of sound through different media

Medium Speed of Sound

(m/s)

Air (0ºC) 331

Air (20ºC) 343

Helium 965

Hydrogen 1284

Water (0ºC) 1402

Water (20ºC) 1482

Aluminum 6420

Steel 5941

Granite 6000 Because temperature can affect the speed of sound, it is generally assumed that ambient temperature—approximately 20ºC—will be the temperature used in testing. The following equation can be used to correct for temperature differences in air. Equation 1:

v ≈ (331 + 0.60T) m/s

Symbol Explanation

v Velocity of sound

T Temperature of air in degrees Celsius

m/s Meters per second units for velocity

≈ Approximately equal (because this equation is not exact)

Figure 2 – A fighter jet breaks the sound barrier and thus produces a shock wave in the skies over the Pacific Ocean. The white halo formed by condensed water droplets is thought to result from a drop in air pressure around the aircraft at speeds approaching the speed of sound.

The frequency of a sound is also known as its pitch. Refer to Figure 1 for an illustration of frequency. The more wavelengths that occur in a given period of time, the higher the frequency, and the shorter the wavelength. As the number of wavelengths increase in a period, the pitch of the sound will be higher. See Figure 3.

Tin-can phones work by converting sound waves into longitudinal mechanical vibrations. As these vibrations are sent down the string, the tension of the string varies. These vibrations eventually reach the bottom of the other can, and the can vibrates. These vibrations are received by the inner ear, and then the brain translates them into language.

Figure 3 – The top row of waves (from a flute) has a high frequency and the sound has a high pitch when received by the ear. The second row of waves (from an oboe) has a lower

frequency and the sound has a lower pitch when received by the ear.

Any detector of sound (such as a human or animal ear, or a microphone) has a characteristic range of frequencies that it can detect. Human hearing can detect frequencies from about 20 Hz to almost 20,000 Hz. This frequency range is referred to as the audible range. Frequencies above 20,000 Hz are called ultrasonic, while frequencies below 20 Hz are called infrasonic. The frequency of sound in any medium is related to the wavelength by the formula:

Equation 2: f = v / λ

Symbol Explanation

v speed of sound in the medium, measured in meters per second

f Frequency of sound measured in Hertz (cycles/second)

λ Greek letter lambda; the wavelength of the sound in the medium, measured in meters

The cracking sound of a bullwhip is a small sonic boom. The end of the whip – known as the cracker – moves faster than the speed of sound, thus creating a sonic boom. The whip is probably the first human invention to break the sound barrier.

Every object has a natural frequency at which it vibrates which is determined by its physical properties. This frequency is the object’s resonant frequency. When an object is exposed to waves that match or are very close to its resonant frequency, it will oscillate with increased amplitude. This effect is called resonance. External forces such as earthquakes or musical instruments can cause the intensity of a material’s vibrations to increase. For instance, a trumpet can release sound waves that force a glass to vibrate. If these vibrations become large enough, the amplitude in the glass molecules can increase to a point that exceeds the plastic limit of the glass, and it will shatter.

Figure 4 – Earthquakes can send vibrations that increase the amplitude of building and structure oscillations. If the resulting increase in amplitude exceeds the plasticity of the building material, the structure will collapse. This roadway was collapsed by the 1989 Loma Prieta earthquake. (USGS)

When sound waves with the same frequency approach each other from opposite directions, they interfere with each other. When these waves overlap, they form standing waves. Figure 5 shows an illustration of a standing wave. Note that the areas with the largest amplitude are called antinodes and the areas with the smallest amplitude are called nodes. Note also that two oscillations equal one wavelength as denoted by the bracket labeled “1.” When the waves overlap in such a manner that resonance occurs, this causes the amplitude to increase, and thus results in a louder sound.

This resonance property will be used to determine the length of the wavelengths of sounds in this lab. A sound wave will be created by vibrating a tuning fork placed over the opening of a plastic (PVC) pipe. The sound waves will enter the pipe, hit the closed end, and bounce back to return to the opening. Because the frequency of the starting and returning waves will be the same, a standing wave similar to the one shown in Figure 5 will form.

Figure 5 — Standing wave

Item Description

1 the length of 1 wavelength λ

2 the length of ¾ λ

3 the length of ¼ λ

* Antinode

+ Node

As a plastic pipe partially submerged into a glass of water is raised and lowered, the ear detects the overlap of waves in a state of resonance as a louder sound coming from the pipe. The closed end of the tube contains a node, which is the smallest amplitude of the wave, and the open end contains an antinode, which is the largest amplitude of the wave. Note in Figure 5 that the distance between any node and its adjacent antinode is λ/4.

NOTE: Although the open end of the pipe provides the proper conditions for an antinode, the actual antinode has been found to occur outside the tube at a distance of about 0.3d from the end, where d is the inner tube diameter. (This end correction may be used to obtain a more accurate value if only one resonance can be measured.)

The plastic tubing provided for this experiment will allow only the λ/4 measurement as the wavelength of the sound emitted by the tuning fork is approximately 1 m in length. In order to measure a full standing wave, a 1-m pipe would be required. If you have access to longer tubing, repeat the following procedure with this tubing and ask the instructor if extra credit can be received for completing this extra experiment.

Exercise 1 You find the wavelength of a certain frequency of sound emitting from a tuning fork. PROCEDURE Assume that this experiment is tested at an ambient temperature of 20oC, so the velocity for sound in traveling in air is 343 m/s. 1. Prepare a data table similar to Data Table 1 shown below. Record observations in Data Table 1: Data Table 1 –Calculation of sound wavelength

Tuning fork frequency (f), Hz (printed on the fork)

Length, L (water level

to top of tube)

Diameter of tube, d

λ = 4(L + 0.3d)

Experimental v = f λ

Room Temperature,

2. Fill a water glass, approximately 17 cm high, with tap water. The glass needs to be wide

enough to hold the plastic pipe.

3. Allow the tap water to equilibrate to ambient temperature. Use a thermometer to measure both the room temperature and the water; when they are the same temperature the water is equilibrated. Record this temperature in Data Table 1.

4. Place the plastic tube into the water. 5. Hold the tuning fork by its handle and strike it against a wooden block or against the

heel of your shoe. 6. Hold the vibrating tuning fork so that the tines are horizontally aligned near the top of the

tube, but not touching the tube. See Figure 6.

Figure 6 — Placement of tuning fork over the open end of the plastic pipe

7. Move the tube slowly up and down in the water until the sound increases. At this point

the wavelengths are resonating. 8. At the point of the loudest sound, hold the pipe in place and use the tape measure to

measure the distance from the top of the plastic tube to the top of the water. It might be helpful to have a partner measure this distance while you hold the pipe in place.

9. Record this length L in meters in Data Table 1. 10. Measure the inner diameter of the resonance tube. Record this measurement d in

meters in Data Table 1. 11. Use the equation: λ = 4(L + 0.3d), to solve for the wavelength of sound. Record this

value in Data Table 1. 12. The frequency of the sound of the tuning fork is indicated on the fork. Record this

frequency in Data Table 1.

Calculations: 1. Calculate the experimental speed of sound:

v = f λ

Symbol Explanation

v Velocity of sound

f Frequency of the tuning fork

λ 4(L + 0.3d)

2. Calculate the theoretical speed of sound (v):

v = 331.4 + 0.6TC m/s

 331.4 m/s is the speed of sound at 20ºC  TC is the temperature of air during testing

3. Calculate the percent error of your experimentally derived value:

% error = experimental value – theoretical value × 100 theoretical value

4. Considering the length of your resonance tube, what is the lowest frequency tuning fork

you could use for this experiment? Show your calculations!

5. A person fishing from a pier observes that four wave crests pass by in 7.0 s and estimates that the distance between two successive crests is 4.0 m. The timing starts with the first crest and ends with the fourth. What is the speed of the wave?

QUESTIONS A. Search for examples of resonance frequencies that have occurred on bridges, sport

stadiums, and mosh pits. Write a one page paper on how these instances of resonance occurred and what effects they had on these structures.

B. Referring to the question above, what steps can be taken in designing a structure to prevent resonant frequencies from occurring. Hint: search the Web for “damping building techniques.”

APPENDIX

Using Statistics This short introduction to statistics is designed to assist you in performing a few simple statistical analyses on some of your experiments. This brief introduction to statistics only scratches the surface. If you want to learn more about statistics, consider reviewing a statistics textbook, downloading one of the many statistics tutorials on the Internet, or taking a statistics course. The final section will show you how to use Microsoft® Excel® to calculate most statistics automatically. Statistics is a branch of applied mathematics. It specifically deals with the collection and interpretation of quantitative data and the use of probability theory to estimate population parameters. Statistics can be categorized into two sub-groups: descriptive statistics and inferential statistics. Descriptive statistics describe large amounts of data in an abbreviated form. They describe important characteristics of the data, including the mean, median, range, variance, standard deviation, etc. Inferential statistics use data obtained from a small group of elements called the sample to make estimates and test hypotheses about the characteristics of a larger group of elements called the population. Descriptive Statistics There are a number of measures of central tendency used to describe the center of a distribution and the scatter of observations around the center.

 Mean: The mean is the arithmetic average of all observations in a distribution. The mean is equal to the sum of all observations divided by the sample size.

 Mode: The mode is the most common value or class in the distribution.

 Median: If all of the observations are arranged in rank order from smallest to largest,

the median is the value bound by 50% of the observations on each side. If the number of observations in the distribution is odd, the median is simply the middle value in the ranked observations. If the number of observations is even, then the median is the mean of the two most central observations.

The measurement of the scatter of observations around the center of a distribution is extremely important. What if you had two means from two populations and the means of both were 50, but the values going into sample #1 were 1, 50, 100 and the values going into sample #2 were 49, 50, 51. It would appear that these are two very different distributions, but with the same mean.

The scatter of observations around the center of a distribution can be depicted in the following ways:

 Range: The range of a distribution is the difference between the largest and the smallest value and is typically expressed as range = 1 – 22, meaning the lowest value was 1 and the highest was 22.

 Variance and standard deviation: The variance is a measure of the average squared

deviation from the mean. Variance differs from the range in that the variance takes into account the distribution of all data points; the range simply describes the single lowest and highest extremes. To calculate variance, take the deviation (or differences) of each value xi (i.e., the ith value of x) from the mean, (i.e., Xi - ). Square these differences and divide by the number of values minus one (i.e., n-1).

The standard deviation (s) is the square root of the variance. The advantage of the standard deviation is that if the data conform to a normal distribution, 95% of the values will fall within two standard deviations (actually 1.96s) on either side of the mean.

If you were contrasting the weight of two populations of acorns, it might be nice to see a statement like" "Acorns from Plot A (mean = 4.58 g, S.D. = .59, range = 3.99 – 4.91) were heavier than the acorns from Plot B (mean = 3.64, S.D. = .71, range = 2.99 – 4.29). At this point you could also give the results of a statistical test comparing these two samples. See the upcoming section on Hypothesis Testing.

Inferential Statistics There are two types of statistical inferences: hypothesis testing and estimation of population parameters. Hypothesis testing refers to a general class of procedures for weighing the strength of statistical evidence or for determining whether the evidence supporting one hypothesis over another is sufficiently strong. Hypothesis testing is one of the most important tools statistical applications bring to real-life problems. Most often, decisions are required concerning populations on the basis of sample information. Statistical tests are used in arriving at these decisions. There are five ingredients to any statistical test:

1. Null Hypothesis 2. Alternate Hypothesis 3. Test Statistic 4. Rejection/Critical Region 5. Conclusion

Following is a simple example of hypothesis testing and the application of a null hypothesis and an alternative hypothesis.

One may wish to test whether or not a coin is fair (that is, whether there is an equal chance of it coming up heads or tails when tossed). The null hypothesis is that the coin is fair; the alternative hypothesis is that the coin is biased. If a series of coin tosses produce a result that is only 4% likely given a fair coin, one would reject the null hypothesis, assuming 95% confidence is required. By contrast, if the experiment produces a result that is 30% likely given a fair coin, one would fail to reject the null hypothesis that the coin is fair. It is not permissible to accept the alternative hypothesis. Only acceptance or failure to reject the null hypothesis is allowed in hypothesis testing. If a test fails to reject the null hypothesis, it is said to lack sufficient power to accept the alternative hypothesis.

The null hypothesis states that there is no effect or difference between procedures and is denoted by H0. The objective of hypothesis testing is to either accept or reject the null hypothesis The alternative hypothesis states that there is a statistically significant difference in the outcome of an experimental procedure. Typically, the null hypothesis is stated first, followed by the alternative hypothesis (Ha). This alternative can be stated simply as there is a true difference. Only one of the two statistical hypotheses can be true.

Consider a simple example in which one wishes to compare the size of male and female fish. The null hypothesis might be that the males and females are the same size (i.e., the samples obtained were drawn from the same underlying population). The alternative hypothesis is that males and females are different sizes. The null hypothesis is tested with an appropriate statistic. If one rejects the null hypothesis, one is left with the alternative that there is a difference (i.e., males and females are of different sizes). This sounds pretty simple, but statistical tests provide a formal means to tell us if the evidence is sufficiently compelling to reject the null and decide that something is going on and might be worthy of further investigation.

Hypotheses can be directional (e.g., males are smaller than females) or non-directional (e.g., males and females are of different sizes), and this determines whether to use what is called a one-tailed or a two-tailed test.

 Example: Two-Tailed Hypothesis Ho = There is no difference in size between male and female fish. Ha = There is a difference in size between male and female fish.

 Example: One-Tailed Hypothesis

Ho = Male fish are not smaller than female fish. Ha = Male fish are smaller than female fish.

In the example for a one-tailed test, failure to reject the null hypothesis might mean that there was no difference in size of male and female fish or that female fish were bigger than male fish. Decision Making and the Level of Significance After stating the hypothesis, one must select and carry out an appropriate statistical test. Each test is based upon a different test statistic [e.g., given the symbols, t (for a t-test), F (in an analysis of variance), r (for a correlation analysis), χ2 (for a chi-square test), etc.]. By plugging the values from a sample into a formula for the statistical test, one ends up with an observed value for the test statistic. One must then compare the observed value of the test statistic with a theoretical distribution of values that one would obtain if the null hypothesis was true. This distribution of expected values is generated from the assumptions that underlie the test and, in the case of parametric tests, from some of the data that was collected, such as the variance among multiple observations within a group. These distributions are typically summarized in tables that are published in statistics books or are readily available on the Internet (search: Statistical Tables).

With these tables, one can ask how likely is it that we would have obtained the observed results. The table provides the entire distribution and determines how much of the distribution lies beyond the observed value of the test statistic. This yields the P-value, or the probability of obtaining one’s observed results or something more extreme under the assumption that the null hypothesis is correct. For example, a P-value of 0.13 means that if the null hypothesis were true, 13% of all possible samples would lead to results as extreme as those found (i.e., with the same or more extreme differences between the two groups). The smaller the P-value, the less likely it is that the null hypothesis is true. But how small should the P-value be before one rejects the null hypothesis? This cut-off is given the symbol α (alpha), which by convention is typically set at 0.05. In other words, 5% of the times when the null hypothesis is correct, one will conclude that the null hypothesis is wrong. This is called a Type I error. The probability of a Type I error is equal to α. To interpret the results, one compares the P-value to α. If P<α (i.e., if P<0.05) then one rejects the null hypothesis. If P> α or P= α, one tentatively accepts the null, recognizing that it might be wrong, but there is insufficient evidence to reject it. If able to calculate a P-value exactly from a distribution or from a statistics software program, it's useful to report the exact value as P=0.05 rather than P<0.05 or P>0.05. In the second and more common case, the entire distribution is not published, so one cannot exactly determine the P-value. Instead, the tables provide particular values of the test statistic associated with different P- values or levels of significance or α. If one’s observed test statistic is greater than this critical value of the test statistic, one can reject the null hypothesis because P < α.

The T-Test One of the most common comparative statistical tests is the t-test. Also called the student’s t-test, it is used when there are just two sets of normally-distributed data to compare. Normally-distributed data means that the data distribution looks like a bell-shaped curve. There are several types of t-tests, each designed mathematically for a specific application. Here we will look at the t-test used to compare two independent samples, which one would use in an experiment where the average height of plants in the two squares sampled are compared. In this exercise you will ask if the difference between the mean heights of the plants in the two plots is statistically significant. Your null hypothesis is that the species of plants present and the conditions in which they have grown have made no difference in their height and that the mean heights of the two plots are essentially the same, allowing for some minor variance:

Ho: µ1 = µ2 The alternative hypothesis is that differences in the plant species and growing conditions have made a difference and that the mean heights are not the same:

Ha: µ1 ≠ µ2

If the mean heights are not the same, then the question is whether the mean height of one sample is significantly larger or smaller than the other. As you have two means, you will use a two-tailed test. Don’t worry if this sounds confusing. Following are step-by-step examples of how this analysis is performed. Sample Problem 1: Weights of acorns collected from two different plots 1. Calculate the mean (average) of the weights in grams. Add all data point values for each

plot and divide by the number of data points.

 Plot A: (2.33 + 2.51 +2.12 +2.7 +2 +2.42 +2.54 +2.6 +2.44 +2.53)/10 = 2.419

 Plot B: (2.02 +1.9 +2.13 +2.5 +2.3 +2.5 +2.3 +2.21 +2.21

+1.8 +2.64+2.14)/10 = 2.185

2. Calculate the variance (s²) of each plot:

a. Square each data value and enter it in a data table: 2.33² = 5.4289, etc.

b. Add all the data values in the last row: 2.33+2.51+2.12 +……= 24.19.

c. Add all the squared data values: 5.4289 + 6.3001 + ------- = 58.94 (rounded).

3. Enter the values calculated in Step 2 into the equation:

Plot A Plot B 2.33 2.02 2.52 1.90 2.23 2.13 2.70 2.50 2.00 2.30 2.42 2.21 2.54 2.21 2.60 1.80 2.44 2.64 2.53 2.14

Plot A Plot B X x2 x x2 2.33 5.4289 2.02 4.0804 2.51 6.3001 1.9 3.61 2.12 4.4944 2.13 4.5369 2.7 7.29 2.5 6.25 2.0 4.0 2.3 5.29 2.42 5.8564 2.21 4.8841 2.54 6.4516 2.21 4.8841 2.6 6.76 1.8 3.24 2.44 5.9536 2.64 6.9696 2.53 6.4009 2.14 4.5796 24.19 58.9359 21.85 48.3247

Plot A: s2 = {58.94 – [(24.19*24.19)/10]/9} = 0.047 Plot B: s2 = {48.32 – [(21.85*21.85)/10]/9} = 0.065

4. Calculate the t-value.

a. Calculate the numerator of the equations above: . Subtract the mean of plot B from the mean of plot A: 2.419 - 2.185 = 0.234.

b. Multiply the difference of the means (0.234) by the √ n (for the sample in the

example n=10, and the √10 is 3.162) which = 0.74.

c. Calculate the denominator: Take the square root of the sum of the two variances calculated earlier: √(0.047 + 0.065) = 0.334.

d. Divide the numerator by the denominator: 0.74/0.334 = 2.24 = t.

e. The calculated t-value is 2.24.

5. Now look at the table of critical values for t and compare the values in the table to your

calculated t.

In order to use the critical values table, you need alpha (α) and degrees of freedom (df). The total number of data points is n, in your case 20 acorns. For a t-test involving two independent means, df = n – 2. In your case, n = 20 so df = 20 – 2 = 18.

Alpha refers to the degree of confidence. The degree needed to accept the null hypothesis is normally 5% or 0.05. Since you are using a two-tailed test, your alpha has to be split between the two tails, giving an alpha of 0.025 for each tail. Go to the table and look under 0.025 and 18 df. You can find such tables on the Internet with a search of t-test critical values.

From the table you see that the value under α = 0.025 and df = 18 is approximately 2.1 Interpretation of the results: Since the calculated t-value of 2.24 is greater than the critical value of 2.1, accept the null hypothesis that the two means are statistically equal. It indicates that the difference between the means is insignificant at the 95% level (100% minus alpha). In other words, the means of the two samples differ by less than can be accounted for by minor variations and the size of the sample. Sample Problem 2: Weights of acorns collected from two different plots with different number of data points in each plot. 1. Calculate the mean (average) of the weights in grams. Add all data point values for each

plot and divide by the number of data points. Plot A: (2.33 + 2.51 +2.12 +2.7 +2 +2.42 +2.54 +2.6 +2.44 +2.53+2.5+2.55)/12 = 2.437 Plot B: (2.02 +1.9 +2.13 +2.5 +2.3 +2.5 +2.3 +2.21 +2.21 +1.8 +2.64+2.14)/10 = 2.185

2. Calculate the variance (s²) of each plot:

a. Square each data value and enter it in a data table, Ex: 2.33² = 5.4289, etc. b. Add all the data values: 2.33+2.51+2.12 +……= 29.24 c. Add all the squared data values: 5.4289 + 6.3001 + ------- = 71.69 (rounded).

Sample 1 Sample 2 2.33 2.02 2.51 1.9 2.12 2.13 2.7 2.5 2.0 2.3 2.42 2.21 2.54 2.21 2.6 1.8 2.44 2.64 2.53 2.14 2.5 2.55

3. Enter the values calculated in Step 2 into the equation:

Plot A: s2 = {71.69 – [(29.24*29.24)/12]/10} = 0.04 Plot B: s2 = {48.32 – [(21.85*21.85)/10]/9} = 0.065

4. An additional step is needed to calculate pooled variance since you have an unequal number of data points in each plot.

sp2 = (n1-1)s12 + (n2-1)s22 n1 + n2 – 2

So, what does this mean?

 n1-1 = number of data points in plot A minus 1  s12 = variance for plot A  n2-1 = number of data points in plot B minus 1  s22 = variance for plot B  n1 + n2 – 2 = number of data points in plot A + plot B minus 2 (which also = df)

sp2 = (11)0.04 + (9)0.065 = 1.025 = 0.051

12 + 10 – 2 20

5. Now, having adjusted the variance for different sample sizes, you can calculate the t- value using a slightly different equation.

t(pooled) =

= 2.437 – 2.185 = 0.2545

= 0.051/12 + 0.051/10 = .00935

√.00935 = 0.0967

Final step: 0.2545/0.0967 = 2.63 calculated t = 2.63

6. Now find the critical t-value in the table. Using alpha = 0.05/2 = 0.025 and df = 22 – 2 =

20, ee find that tcritical = 2.086. Interpretation of the results: Since the calculated t-value of 2.63 is greater than the critical value of 2.086 you reject the null hypothesis that the two means are equal. It indicates that the difference between the means is significant at the 95% level (100% minus alpha).

The Chi-Square Test The chi-square (X2) test is one of the most useful non-parametric statistical tests for the biologist. It is used with count data or frequencies organized in a matrix defined by two or more variables. The chi-square test is based on the differences between the observed results and the expected values (those results that would be obtained if the null hypothesis were true). The formula for X2 is as follows:

where o is the observed frequency and e is the frequency expected under the null hypothesis of no difference between groups.

Example: Suppose a fisheries biologist samples adult fish from two lake populations: 100 from Lake 1 and 150 from Lake 2. The biologist records whether or not the lakes are infested with a nematode parasite that encysts in their muscles. The biologist wants to know whether the presence of the parasite is independent of the lake from which they were taken. i. Arrange the data in a data table. ii. Calculate the sums for each table row and column.

Data Table 1 – Observed values Site # fish w/parasites #fish w/out parasites Total Lake A 15 85 100 Lake B 50 100 150 Totals 65 185 250 iii. Compute the table of expected values. For example, the expected value for the number

of fish with parasites in Lake 1 = (100x65)/250=26 [i.e., (the row total x the column total)/total].

Data Table 2 – Expected values Site # fish w/parasites #fish w/out parasites Total Lake A 26 74 100 Lake B 39 111 150 Totals 65 185 250

Notice the row and column totals are the same in the tables of expected and observed values.

iv. Compare the observed and expected frequencies using the �2 statistic.

�2= (15-26)2/26 + (85-74)2/74 + (50-39)2/39 + (100-111)2/111 = 10.5 v. Determine the degrees of freedom for the test = (2 rows-1) x (2 columns-1) = 1 df. vi. Compare the calculated �2 value (10.5) with the value for 1 degree of freedom from a

stats table. Since your calculated value is greater than 3.84 (from the table), you can reject the null hypothesis that the presence of parasites in fish is independent of the lake.

Table 2 – Chi-square

Microsoft Excel and Statistics Microsoft Excel spreadsheets can be used to perform most statistical calculations. Analysis ToolPak add-in MS Excel provides a set of data analysis tools, the Analysis ToolPak, which can save steps when one is developing complex statistical or engineering analyses. One provides the data and parameters for each analysis, and the tool uses the appropriate statistical or engineering macro functions before displaying the results in an output table. Some tools generate charts in addition to output tables. To view a list of available analysis tools, go to the Tools > Data Analysis. If the Data Analysis menu item is missing from the Tools menu, the Analysis ToolPak must be installed. Install and Use the Analysis ToolPak 1. Select Tools > Add-Ins > Analysis ToolPak.

If the Analysis ToolPak is not listed in the Add-Ins dialog box, click Browse and locate the drive, folder name. The file name for the Analysis ToolPak add-in, Analys32.xll, is usually located in the Microsoft Office\Office\Library\Analysis folder Run the Setup program if the pak isn't installed.

Excel provides many statistical, financial, and engineering worksheet functions. Some of the statistical functions are built-in and others become available when you install the Analysis ToolPak. It is easy to analyze data using descriptive statistics in Excel, because Excel includes all common statistics, including mean, median, mode, standard deviation, etc. Create a Data File 1. Open Excel. 2. Enter the data at right. 3. Click once in cell C1 to the right of

the Sample 2 column.

4. Click Tools > Data Analysis. The

screen at right will appear.

5. Select Descriptive Statistics.

6. Click OK.

7. Enter the cell reference for the

range of data to analyze. The reference must consist of two or more adjacent ranges of data arranged in columns or rows.

Choose the values for Sample 1 by entering the range, $A$1:$A$11, into the Input Range textbox.

You can also click the drop-down box in the Input Range field and drag the mouse across the desired range.

8. Indicate whether the data in the

input range is arranged in rows or columns by selecting the correct option button. For this exercise, select Columns

9. Indicate whether the input range

contains labels by selecting the Labels in First Column checkbox.

If the input range has no labels, leave the checkbox unmarked. Excel will generate appropriate data labels for the output table.

10. If you want to include a row in the output table for the confidence level of the

mean:

Enter the desired confidence level in the text box.

For example, entering a value of 95% calculates the confidence level of the mean at a significance of 5%.

11. If you want to include a row in the output table for the kth largest value for each

range of data, enter the number to use for k. If you enter 1, this row contains the maximum of the data set.

12. If you want to include a row in the output table for the kth smallest value for each range of data, enter the number to use for k. If you enter 1, this row contains the minimum of the data set.

13. Output Range: Enter the reference for the upper-left cell of the output table.

This tool produces two columns of information for each data set. The left column contains statistics labels, and the right column contains the statistics.

Excel writes a two-column table of statistics for each column or row in the input range, depending on the Grouped By option selected. If you don't enter an output range, Excel might overwrite your data table.

For this example, enter $C$1 in the Output Range text box.

14. Click OK.

After some calculation time, a table will appear in your spreadsheet with all your descriptive statistics.

Perform a t-Test Analysis This tool is a part of the Analysis ToolPak. When two populations are both normally or approximately normally distributed, and when at least one sample size is small (less than 30), the t-test is used to make decisions about differences between the population means. The Analysis ToolPak provides three tools that can be used to test the means of different types of populations. There are four different ways of doing a t-test in Excel.

 t-test: One Sample t-test  t-test: Two-Sample Assuming Equal Variances Analysis  t-test: Two-Sample Assuming Unequal Variances Analysis  t-test: Paired Two Sample for Means Analysis

Single Sample t-Test  Definition: Used to compare the mean of a sample to a known number (often 0).  Assumptions: Subjects are randomly drawn from a population, and the distribution of the

mean being tested is normal.  Test: The hypotheses for a single sample t-test are:

o Ho: u = u0 o Ha: u < > u0

where u0 denotes the hypothesized value to which you are comparing a population mean.

 Test statistic: The test statistic, t, has N-1 degrees of freedom, where N is the number of

observations.  Results of the t-test: If the p-value associated with the t-test is small (usually set at p <

0.05), there is evidence to reject the null hypothesis in favor of the alternative. In other words, there is evidence that the mean is significantly different than the hypothesized value. If the p-value associated with the t-test is not small (p > 0.05), there is not enough evidence to reject the null hypothesis, and the conclusion is that there is evidence that the mean is not different from the hypothesized value.

To create a single sample t-test in Excel, you use the tdist function. Example: TDIST (1.96,60,2) equals 0.054645, or 5.46 percent Exercise - Single Sample t-test from http://www.mastep.sjsu.edu/learn/t_test.htm You have been told that the average employee for your industry has an average dexterity score of 100 on a standardized test. You think your employees will score differently, so you give a random sample of 12 the test. The results are: First, we need to construct hypotheses.

Ho: The average dexterity score for our employees is 100. Ha: The average dexterity score for employees is not 100.

1. Open an Excel® spreadsheet and enter the data

from the table.

Subj. Test Score

1 98

2 102

3 120

4 140

5 123

6 101

7 89

8 99

9 119

10 103

11 132

12 107 2. Move the cursor to cell E4. 3. Calculate the sample mean and the standard deviation of our sample. 4. Click Insert > Function.

5. In the left column, click Statistical.

You will see a list of statistical functions appear on the right.

6. Scroll down until you see AVERAGE.

7. Click AVERAGE

8. Either type the range B2:B13

into the Number1 box or use the mouse to select the range from the worksheet.

9. Click OK.

For this example, you should get a mean of 111.0833..

10. Next, click cell E3. 11. Click Insert > Function.

12. In the left column, click Statistical.

You will see a list of statistical functions appear on the right.

13. Scroll down until you see STDEV.

14. Click STDEV.

15. Either type the range B2:B13

into the Number1 box or use the mouse to select the range from the worksheet.

16. Click on OK.

For this example you should get a standard deviation of 15.45938. We are using the standard deviation of a sample [with n-1] because we know it is a sample, not the entire population.

17. Calculate the t ratio using the following formula.

t = Sample Mean – Population Mean Sample SD / (SQRT (sample size))

For our example, this calculation would be t = 111.0833 - 100 15.45938/(SQRT(12))

t = 2.4835

18. To test the hypothesis that our sample is different from the population, we need to find

the two-tailed probability of a t ratio of 2.4835 with 11 degrees of freedom Remember degrees of freedom is Sample Size - 1.

19. Next, click cell E4. 20. Click Insert > Function. 21. In the left column, click Statistical.

You will see a list of statistical functions appear on the right.

22. Scroll down until you see TDIST.

23. Click TDIST.

24. Enter the values at right into the text boxes.

25. Click OK.

You should get a value of 0.030384. This means there is a 3% chance that your sample is representative of the population.

Related to significance levels, you would reject the null hypothesis at the 5% significance level, but you would not reject the null hypothesis at the 1% level. About the t-Test: Two-Sample Assuming Equal Variances Two samples are referred to as independent if the observations in one sample are not in any way related to the observations in the other. This is also used in cases where you randomly assign subjects to two groups, give the first group treatment 1, give the second group treatment 2, and compare the two groups. This analysis tool performs a two-sample student's t-test. This t-test form assumes that the means of both data sets are equal; it is referred to as a homoscedastic t-test. You can use t- tests to determine whether two sample means are equal. For this example, you are going to use the data from the file space.xls. In a NASA-funded study, 7 men and 8 women spent 24 days in seclusion to study the effects of gravity on circulation. Without gravity, there is a loss of blood from the legs to the upper parts of the body. The study started with a 9-day control period in which the subjects were allowed to walk around. This was followed by a 10-day bed rest period in which the subjects' feet were

somewhat elevated to simulate weightlessness. The study ended with a 5-day recovery period in which the subjects were allowed to walk around. Every few days, researchers measured the electrical resistance at the calf, which increases when there is blood loss. The electrical resistance gives an indirect measure of the blood loss. Example from Data Analysis with MS Excel®, p. 137. 1. Open Excel.

2. Enter the data set at right.

3. We are going to use Excel's Analysis ToolPak to perform a 2-sample t-test. Click Tools > Data Analysis.

4. Scroll down until you see t-test: Two-Sample Assuming Equal Variances.

5. Click on this t-test. 6. Define the Variable 1 Range.

This is the cell reference for the first range of data you want to analyze. The range must consist of a single column or row of data. In this case, we are going to select the range $A$2:$A$12. You can either type this range or use the mouse to select the range from the worksheet.

7. Select the Variable 2 Range. We are going to select the range $B2:$B$12.

8. Hypothesized Mean Difference: Enter the number that you want for the shift in sample

means. A value of 0 (zero) indicates that the sample means are hypothesized to be equal.

9. Enter 0. 10. Labels: Select the checkbox if the first row or column of your input ranges contains

labels.

Clear this check box if your input ranges have no labels. Excel generates appropriate data labels for the output table. Since your first row contains labels (Sample 1 and Sample 2), check the Labels box.

11. Alpha: Enter the confidence level for the test. This value must be in the range 0 to 1.

The alpha level is a significance level related to the probability of having a type I error (rejecting a true hypothesis). We will enter 0.05 for our Alpha.

12. Output Options: You have three options

for the output range. Generally, it is safer to choose either New Worksheet Ply or New Workbook.

Output Range: If you place a range here, your t-test values will appear on the same worksheet.

Enter the reference for the upper-left cell of the output table. Excel automatically determines the size of the output area and displays a message if the output table will replace existing data.

New Worksheet Ply: Select to insert a new worksheet in the current workbook and paste the results, starting at cell A1 of the new worksheet.

To name the new worksheet, type a name in the box.

New Workbook: Select to create a new workbook and paste the results on a new worksheet in the new workbook.

For this example, select Output Range A17. This means the results will be displayed below the data table.

Click OK. You should see the information at right in a new worksheet.

Let's translate this data into plain English. Based on the table, we can see that, on the average, the sample 1 acorns have higher weight (2.419g as compared to 2185 g for sample 2 acorns). The variances in the two samples are not exactly the same, so we may want to recheck our calculations using the t-test with unequal variances option in Excel. Generally, if the variance is close, the difference in t value will not be great. The value of the t statistic is 2.217. The two-tailed p-value is given as 0.0397, which is less than 0.05, so the difference is significant at the 5% level. Because the probability level is so small, the null hypothesis of no difference between sample 1 and sample 2 seems incompatible with the data. Therefore, you would reject the null hypothesis and state that the weights are different for sample 1 and sample 2 acorns.

About the t-Test: Two-Sample Assuming Unequal Variances One would use this test if the variances in the two groups are extremely different. The worst situation would be if the two samples are of very different sizes, and the small sample has a much larger standard deviation. We will use the same data file as above, Space.xls, to do this calculation.

1. Open Excel and use the same acorn data as before. 2. Click Tools > Data Analysis. 3. Scroll down until you see t-test: Two-

Sample Assuming Unequal Variances. 4. Click once on this t-test.

5. Enter the same data as you did in the t-

test: Two Sample Assuming Equal Variances example.

6. Click OK. You should see the information at right in a new worksheet.

As you can see, the values are identical to those for equal variances, and our assumption about equal variances was correct. If the variances had been unequal, we might have gotten slightly different results. But as long as the standard deviations and the sample sizes are close, the results will often be very close to those of the equal variances t-test. t-Test: Paired Two Sample For Means Analysis This analysis tool and its formula perform a paired two-sample student's t-test to determine whether a sample's means are distinct. This t-test form does not assume that the variances of both populations are equal. You can use a paired test when there is a natural pairing of observations in the samples, such as when a sample group is tested twice — before and after an experiment. Another reason data is dependent is when results on one measure are presumed to be related to another measure. For example, if a student does well in one subject, English, it is likely that he will do well in another subject, for example, history. In fact, this is the situation we are going to use to demonstrate this t-test.

Paired t-test

 Definition: Used to compare means on the same or related subject over time or in differing circumstances.

 Assumptions: The observed data are from the same subject or from a matched

subject and are drawn from a population with a normal distribution.

 Characteristics: Subjects are often tested in a before-after situation (across time, with some intervention occurring such as a diet), or subjects are paired such as with twins, or with subject as alike as possible. An extension of this test is the repeated measure ANOVA.

 Test: The paired t-test is actually a test that shows the differences between the two

observations is 0. So, if D represents the difference between observations, the hypotheses are: Ho: D = 0 (the difference between the two observations is 0) Ha: D 0 (the difference is not 0)

The test statistic is t with n-1 degrees of freedom. If the p-value associated with t is low (< 0.05), there is evidence to reject the null hypothesis. Thus, you would have evidence that there is a difference in means across the paired observations. Sample Calculation: We want to find out whether a student's performance in English is, on the average, different from his/her performance in history. Suppose that a sample of 11 students is selected and their grades for these two subjects are obtained. 1. Open Excel and enter the data at right.

2. Click Tools > Data Analysis. 3. Scroll down until you see t-Test: Paired

Two-Sample for Means. 4. Click once on this t-test.

5. In this case, we are going to select the range $B$3:$B$14. You can either type this

range or use the mouse to select the range from the worksheet.

6. Next, select the Variable 2 Range, and select the range $C$3:$C$10.

7. Hypothesized Mean Difference: Enter the number that you want for the shift in sample means. A value of 0 (zero) indicates that the sample means are hypothesized to be equal. Enter 0.

8. Labels: Select if the first row or column of your input ranges contains labels.

Clear this check box if your input ranges have no labels; MS Excel generates appropriate data labels for the output table.

Since our first row contains labels (English and History), you should check the Labels box.

9. Alpha: Enter the confidence level for the test. This value must be in the range 0...1.

The alpha level is a significance level related to the probability of having a type I error (rejecting a true hypothesis).

We will enter 0.05 for our Alpha.

10. Output Range: As before, you have three options for the output range. Generally, it is safer to choose either New Worksheet Ply or New Workbook.

Output Range: If you place a range here, your t-test values will appear on the same worksheet. Enter the reference for the upper-left cell of the output table. MS Excel® automatically determines the size of the output area and displays a message if the output table will replace existing data.

New Worksheet Ply: Click to insert a new worksheet in the current workbook and paste the results starting at cell A1 of the new worksheet. To name the new worksheet, type a name in the box.

New Workbook: Click to create a new workbook and paste the results on a new worksheet.

11. Choose New Worksheet Ply and type

the name, paired t-test, to the right. 12. Click OK. You should see the information at right in a new worksheet.