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Phys 1215 Laboratory Manual Spring 2014 1

GENERAL PHYSICS I

MECHANICAL, WAVES AND THERMODYNAMICS

LABORATORY MANUAL

Phys 1215 Laboratory Manual Spring 2014 2

Introduction

As part of this course you will be required to perform various experiments, which are listed within this manual, and plot your results in a manner that is considered “publication quality”. In other words, the figures should be embedded into a word processing document, appropriately labeled and accompanied by a descriptive figure caption. The beginning of the manual will help you with this aspect of the course. In particular, you will be expected to plot your results using Octave. Octave is a free (as in both beer and speech) software package that uses the same syntax as Matlab. Excel is not good enough and, especially in the second semester of physics, incapable of generating the scientific plots required for a scientifically rigorous course. You can get Octave from http://octave.sourceforge.net/ or, if you use Linux, obtain it directly from the repositories (there is also an app for some tablets). Octave is awesome and I'll walk you through how to use it. Being able to plot data is an important part of being a scientist or engineer, and learning how to plot figures will serve you well, not only in other courses that require you to generate plots and figures but, hopefully, in your future careers.

Specifically, the objectives of this course are:

● The course should be light and relatively easy as it is only a 1 credit course .

● You should learn about different types of plots for displaying data (also relevant outside of physics) .

● The experiments might help you understand some of the concepts in class.

● You should have fun and be creative!

The following experiments will make up this course:

I. Uncertainties

II. Acceleration due to gravity

III. Projectile motion

IV. Fractal dimensions in nature

V. Conservation of energy

VI. Pendulum

VII. Atwood's machine

VIII. Hooke's law

IX. Sound waves.

X. Greenhouse Effect

XI. Bernoulli's equation

Only 7 of these 8 Experiments will be completed. All 8 experiments will be set up for 7 weeks and groups will rotate around the room each week.

Note that there are 11 experiments listed, but you are only required to complete 10 experiments during the semester. There is an extra one as a back up. If you want to change the experiment in any way, go for it! Just check with your instructor first. We will meet for 8 weeks, during which time you will conduct the first 8 experiments. Then we will have 3 weeks in which you will conduct your own experiment. The greenhouse effect and Bernoulli equation labs will be conducted during weeks 12 and 13. Group presentations will be in week 14.

You should print out this lab manual (or have electronic access to it during lab times) and purchase a notebook for writing down your measurements (or type them directly into a spreadsheet during the lab). You will also need to have access to Octave (which is a free math and plotting software) which you can download to your own computer.

http://octave.sourceforge.net/

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Plotting Figures and Using Octave

1. Plotting Figures.

Here are the general rules for figures which are published in scientific journals and will be applied to the figures you include in your right-ups for this course.

1. The background color should be white (and not some randomly shaded pattern).

2. The data should be clear and easy to see (i.e., not yellow lines).

3. If more than one line is plotted in the same graph then you should include a legend or key which identifies the different data, and plot the different data with different colors.

4. The plot should include an appropriately labeled x-axis and a y-axis. The appropriate label will consist of the variable being plotted (for example, time or distance) along with the units (seconds or meters). 5. The plot should be of your actual data and usually consist of points, with a line sometimes provided as a numerical fit or as a “guide to the eye”.

6. If Greek letters are required then Greek letters should be included. Not knowing how to do something is no justification for not doing it. Google it or ask me if you don't know how.

7. Combining subfigures are a common way of displaying data. Subfigures can be either inside another figure or below it (or both?). If it is inside another figure it is not usually separately labeled, but rather referred to as “inset”. If it is two figures plotted together then they would commonly be referred to as Figure 1A and 1B, for example.

8. Real plots don't have titles.....

9. ...instead they have “figure captions” which are displayed below the figure. This is a sentence or two which explains the plot. One will often flip through a journal and gain a snapshot, or impression, of the article based on the figures and the figure captions, so these are very important.

10. Last, but certainly not least, your figures should “tell a story”. If nothing can be obtained from reading your figure then why include it? It would be like including a series of meaningless text for a lab report. If your plot doesn't mean anything then don't even bother handing it in!

Look at the following couple of examples to get an idea of what a figure should look like. These figures are both taken from the journal Nature.

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2. Examples.

Figure 1. Submillimetre water emission lines from comet 103P/Hartley 2. The time of the observations was 20 days after perihelion, when the comet was 1.095 AU from the Sun and 0.212 AU from Herschel.

Note that the subfigures are labeled a and b in the figure and should be referred to as Fig. 1a and Fig. 1b in the text. (The capital letter F on Fig. represents its a name and the period is because Fig. was used instead of Figure, which is common.) Here the distinction between Fig. 1a and Fig. 1b was provided by the key (which strictly speaking wasn't necessary) which meant it wasn't necessary in the figure caption. Note also that the units are in parenthesis after the quantity in the axis labels.

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Figure 2: Annual mean difference (open land minus forest) in surface air temperature. a, Correlation with latitude. b, Correlation with surface net radiation. The inset to a has the same axes as the main panel but also shows tropical FLUXNET site data.

Here the subfigures Fig. 2a and 2b have different x-axis so, unlike Fig. 1, do not share the same x-axis. An inset is included in Fig. 2a, but because space is limited the inset doesn't have labels. But we can't just let the poor reader guess what the inset is plotting, so this is mentioned in the figure caption. These plots contain a straight line fit, which you will be required to do also for some of your experiments. Note that the equation which describes the fit is included in the figure. Sometimes the numerical fit can be simply called “numerical fit” in the key, or legend, and the equation given in the text.

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3. Using Octave to Plot Figures.

Lets just say I have some data:

Position (m) Time (s)

10.0 0

12.5 1

14.1 2

15.8 3

18.1 4

20.0 5

I would like to generate a plot of position versus time. To do this I first create vectors or arrays of the position and time values. Note that these must be equal in length! The syntax is:

x=[0 1 2 3 4 5] y=[10 12.5 14.1 15.8 18.1 20]

Rather than type this into octave directly, first type this into notepad and then copy and paste into octave. (Note, to paste into octave right click the bar on top of the window and select edit → paste.) OK lets plot our data:

plot(x,y)

But we have to label the axis, right?

xlabel('Time (s)') ylabel('Position (m)')

This is looking like a real figure! We don't need to include a legend as there is only one set of data, but we should display the actual points rather than just a line, and the data looks kind of straight – how about fitting a straight line to this?

plot(x,y,'o-')

Gives us the points as well as a line, but not a line fit....

p = polyfit(x,y,1); plot(x,polyval(p,x)); hold on; plot(x,y,'k*'); xlabel('Time (s)') ylabel('Position (m)')

Fits a straight line through the data (a polynomial of order 1 hence the use of polyfit with the argument of 1 at the end). We can also use fsolve() which is a function for solving nonlinear equations if our function is messier. Notice the semicolons. These say to Octave “don't output anything to the screen”. However, try typing p....

octave:9> p p =

1.9571 10.1905

… it gives the slope and intercept, respectively!

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However it's unusual that our data would just be numbers, usually we have errors associated with our numbers. More on this later.

Lets redefine our data to include error bars. We must have a lower and upper bound to our error bars, however we'll assume these are the same for now (although Octave can handle these differently). Therefore, our data becomes

Position (m) Time (s)

10.0 .1 0

12.5 .2 1

14.1 .1 2

15.8 .4 3

18.1 .3 4

20.0 .2 5

The errors can be entered into Octave now

x=[0 1 2 3 4 5] y=[10 12.5 14.1 15.8 18.1 20]

ey = [.1 .2 .1 .4 .3 .2]

And to plot this we use the function errorbar() where ey is the error in y. I'm going to create a handler (variable h) for the plot, so that I can later change its properties.

h = errorbar (x, y, ey, "k->") ; xlabel('Time (s)', 'FontSize', 20) ylabel('Position (m)', 'FontSize', 20) set (h, "linewidth", 10) axis('tight') set(gca,'FontSize',20, 'LineWidth', 3)

Notice I've changed the font of the labels to 16 and the axis to 16 also, and made the axis 'tight' rather than start at zero. You can change all of this stuff to get it to look right. Its also worth noting that if you don't like the xlabel or ylabel (sometimes the positioning is silly in my opinion) you can opt to create a separate label and insert it entirely independent of the plot and position it wherever you want.

The most important thing to remember is that if you play around with this stuff then you can get it to look exactly as you want it to. Have a play around and google the different commands and syntax. There are lots of websites explaining the syntax for Octave, which bear in mind is identical to Matlab! There is also a very useful mailing list that you can post problems to (or just come and see me). You can also type help <command> for information about a command.

I'll include the syntax for plotting various data in the lab manual when you have to plot something, which is going to be more crucial in Physics II when you have to plot vectors and surface plots. Most of the plots required for Physics I are covered in this short introduction. You might want to use it in other courses though as its a very powerful (and free) tool!

Oh, I almost forgot, once your done and you like your plot its time to save it, or copy it. In windows, you can right click on the bar across the display window for your plot and either copy (to be subsequently pasted into a word processor) or save it. In linux, type:

print -dpng plot.png

The output is overleaf.

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This is just a quick example, which I think looks fine. Before doing a second plot, I suggest you quit octave (just type “quit”) and re-open it. This will stop commands from your first plot meddling with your second plot. Octave is pretty awesome so try playing around with it. Here's a couple of examples of the advanced functions in Octave.

Try typing:

n = 50; x = y = linspace (-8, 8, n)'; [xx, yy] = meshgrid (x, y); r = sqrt (xx .^ 2 + yy .^ 2) + eps; c = 5 * sin (r) ./ r; h= surf(xx,yy,c,c); shading interp

[x,y] = meshgrid(1:20); u = cos(2*pi*y/10); v = cos(2*pi*x/10); quiver(x,y,u,v,'b',1);

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Uncertainties and Error Propagation

1. Systematic and Random Errors.

No measurement made is ever exact. The accuracy (correctness) and precision (number of significant figures) of a measurement are always limited by the degree of refinement of the apparatus used, by the skill of the observer, and by the basic physics in the experiment. In doing experiments we are trying to establish the best values for certain quantities, or trying to validate a theory. We must also give a range of possible true values based on our limited number of measurements.

Why should repeated measurements of a single quantity give different values? Mistakes on the part of the experimenter are possible, but we do not include these in our discussion. A careful researcher should not make mistakes! (Or at least she or he should recognize them and correct the mistakes.)

We use the synonymous terms uncertainty or error to represent the variation in measured data. Two types of errors are possible. Systematic error is the result of a mis-calibrated device, or a measuring technique which always makes the measured value larger (or smaller) than the “true” value. An example would be using a steel ruler at liquid nitrogen temperature to measure the length of a rod. The ruler will contract at low temperatures and, therefore, consistently overestimate the true length. Careful design of an experiment can allow us to eliminate or to correct for systematic errors.

Even when systematic errors are eliminated there will remain a second type of variation in measured values of a single quantity. These remaining deviations will be classed as random errors, and can be dealt with in a statistical manner. Let's look at how we can quantify the random errors of our measurements.

2. Determining Random Errors.

How can we estimate the uncertainty of a measured quantity? Several approaches can be used, depending on the application.

a) The Least Count

The least count is the smallest division that is marked on the instrument. Thus a meter stick will have a least count of 1.0 mm, or a digital stop watch might have a least count of 0.01 sec. This is just saying that given what instrument you are using then this is the best you can do.

b) Estimated Uncertainty

Often other uncertainties are larger than the least count. We may try to balance a simple beam balance with masses that have a least count of 0.01 grams, but find that we can vary the mass on one pan by as much as 3 grams without seeing a change in the indicator. We could use half of this as the estimated uncertainty, thus getting uncertainty of ±1.5 grams. A lot of the time this is just guess work.

c) Standard Deviation: Estimated Uncertainty by Repeated Measurements

The statistical method for finding a value with its uncertainty is to repeat the measurement several times, find the average, and then find the standard deviation.

Calculating the standard deviation is easy. Put your data in a spreadsheet and use the function STDEV() to calculate it for you. But lets look at what the standard deviation is.

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Standard Deviation

Suppose we repeat a measurement several times and record the different values. We can then find the average value, sometimes denoted by a symbol between angle brackets, <t>, and use it as our best estimate of the reading. How can we determine the uncertainty? Let us use the data below as our example. Column 1 shows a series of measurements of time in seconds.

A simple average of the times is the sum of all values (7.4+8.1+7.9+7.0) divided by the number of readings (4), which is 7.6 sec. We can use angular brackets around a symbol to indicate average; an alternate notation uses a bar placed over the symbol. Note we could use AVERAGE() to calculate this in most spreadsheets.

Column 2 of Table 1 shows the deviation of each time from the average, (t-<t>). A simple average of these is zero, and does not give any new information. To get a non-zero estimate of deviation we take the average of the absolute values of the deviations, as shown in Column 3 of Table 1. We will call this the average deviation, dt.

Table 1. Values showing the determination of average, average deviation, and standard deviation in a measurement of time. Notice that to get a non-zero average deviation we must take the absolute value of the deviation.

Time, t (sec) (t - <t>) dt = | t - <t>| (t - <t>)2

7.4 -0.2 0.2 0.04

8.1 0.5 0.5 0.25

7.9 0.3 0.3 0.09

7 -0.6 0.6 0.36

Average = 7.6s Standard deviation = 0.5

Column 4 has the squares of the deviations from Column 2, making the answers all positive. The sum of the squares is divided by 3, (one less than the number of readings), and the square root is taken to produce the sample standard deviation. An explanation of why we divide by (N-1) rather than N can be found in any statistics text.

Mathematically this is commonly written as

where N is the number of samples, is the standard deviation, xi is the ith reading and is the average. If you use a spreadsheet there are built-in functions that help you to find these quantities, and for the standard deviation you can simply use the STDEV() function to calculate this for you (you just give it your xi's).

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3. What is the Range of Possible Values?

When you see a number reported as (7.6 ± 0.4) sec your first thought might be that all the readings lie between 7.2 sec and 8.0 sec. A quick look at the data in the Table 1 shows that this is not the case. Statistically we expect 68% of the values to lie in the range of , but that 95% lie within .

How do we use the uncertainty? Suppose you measure the density of calcite as (2.65 ± 0.04) g cm -3. The textbook value is 2.71 g cm-3. Do the two values agree? Since the text value is within the range of two deviations from the average value you measure you claim that your value agrees with the text. If you had measured the density to be (2.65 ± 0.01) you might be forced to admit your value may disagree with the text value. Therefore, errors are important.

The drawing below shows a Normal Distribution (also called a Gaussian). The vertical axis represents the fraction of measurements that have a given value x. The most likely value is the average, in this case = 5.5 cm. The standard deviation is = 1.2. The central shaded region is the area under the curve between ( - ) and ( + ), and roughly 67% of the time a measurement will be in this range. The wider shaded region represents ( - 2 ) and ( + 2 ), and 95% of the measurements will be in this range.

4. Relative and Absolute Errors

The quantity z will have units and is called an absolute error while z/z is called a relative error or fractional uncertainty. Percentage error would be the fractional error multiplied by 100%. In practice, either the percentage error or the absolute error may be provided.

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5. Propagation of Errors, Basic Rules

Suppose two measured quantities x and y have uncertainties, and , determined by procedures described in previous sections: we would report ( ), and ( ). From the measured quantities a new quantity, z, is calculated from x and y. What is the uncertainty, , in z? The guiding principle is to consider the most pessimistic situation.

(a) Addition and Subtraction: z = x + y or z = x - y

(b) Multiplication and Division: z = x y or z = x/y

(d) In general:

where z is a function of x, y, etc...

Example

w = (4.52 ± 0.02) cm, x = ( 2.0 ± 0.2) cm, y = (3.0 ± 0.6) cm. Find z = x + y - w and its uncertainty.

z = x + y - w = 2.0 + 3.0 - 4.5 = 0.5 cm

and standard deviation, = 0.633 cm, from the equation above.

Therefore z = (0.5 ± 0.6) cm. Notice that we round the uncertainty to the same significant figures as the answer.

Example

The radius of a circle is x = (3.0 ± 0.2) cm. Find the circumference and its uncertainty.

= 18.850 cm

= 1.257 cm (The factors of 2 and are exact, and have no error).

Therefore, C = (18.8 ± 1.3) cm.

Again we always round the uncertainty to the same number of significant figures as our answer.

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6. Procedure

Measure the following 5 quantities:

1. Average height of several class mates. 2. The area of someones hand. 3. The volume of the physics lab. 4. The temperature difference between inside and outdoors (yes I know its cold outside). 5. The velocity of random people walking through Romo's.

7. Analysis

Calculate the errors associated with these measurements!

8. Grade

Each of the above 5 measurements will be worth 20% of your grade for this laboratory (10% for the measurement and 10% for calculating the appropriate error). Therefore, read through the above section on the propagation of errors very carefully. The laboratory report should be no more than 1 page in length if possible, entirely type written and include the calculation of your errors. Hand written labs will not be graded.

The above equations for propagating errors will be used throughout this lab and Physics 2 lab. A number should never be written on it's own. It must always be accompanied with an error!

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Acceleration Due to Gravity

1. Introduction and Objectives

The gravitational interaction between two bodies of masses m and M is

For the surface of the earth, the radius is that of the earth (6378 km), the mass of the earth is 5.97x1024 kg and the constant G is the universal gravitational constant, 6.67x10-11 m3 kg-1 s-2. Therefore, an object on the surface of the earth experiences a constant acceleration of the form

The purpose of this lab is to investigate this acceleration and to verify that the acceleration is a constant.

2. Apparatus

Balls of different mass Camera connected to a computer (with “guvcview” and “tracker” installed) Meter stick

3. Procedure

1. A uniform background is required and this should be performed next to the white wall at the end of the work bench.

2. The software you will be using requires you to measure distance. Therefore, a meter stick should be placed against the wall and held in place to give your analysis scale. You can tape this to the wall.

3. Position the camera so that it points towards the wall and includes the meter stick in the view. The software “guvcview” can be used to record video from the camera and can be found under “Sound & Video” in the main menu. Make sure that the software is set up correctly (it should be already):

Under “Image Controls” the Exposure, Auto should be set to “Manual Mode” Under “Image Controls” the Exposure (absolute) should be set to around 20 (any less and it looks dark). Under “Videos & Files” the frame rate should be “30/1 fps” Under “Videos & Files” you can also change where the computer will save the videos. Feel free to play around with brightness and contrast, but note that you have to be able to read the markings on the meter stick to give the scene scale.

4. You will be investigating the gravitational acceleration of two different balls with different masses. Select two balls (same volume) and measure their masses using the scale provided.

5. Press record on the software “guvcview” and drop a ball. Press stop to finish taking the video. You should record at least one decent video of each ball falling. To view the video using “vlc”.

6. Repeat the above procedures but with the different ball (with different mass).

You should end up with values for the two masses (as measured using the scale) and 2 videos (for the 2 different balls).

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4. Analysis

You should have 2 video files, capturing the two different balls as they fell. You will analyze their acceleration due to gravity using a software called “tracker”. The software is on the laptop provided as well as available to download:

http://www.cabrillo.edu/~dbrown/tracker/

To analyze your videos, follow the procedures below

1. “Tracker” doesn't seem to like the avi file format and prefers mov format instead. Therefore, we'll convert your two chosen files. You can convert in vlc or using the command line:

Here it is assumed that the videos are in the Desktop folder (although, you may have saved them in the home folder) From the main menu open up a terminal Accesories→LXTerminal then type

cd Desktop avconv -i “capture-1.avi” -r 30 “output1.mov”

where the “capture-1.avi” is the name of your file and “output1.mov” is the name of the file your creating.

2. Open the “Tracker” software. It can be found under “Education” in the main menu.

3. Click on “Video→Import” and then go to the video (should be in mov format) you want to analyze.

4. You can step through the video, frame by frame, by pressing the button.

5. Go to the frame just as the ball is on free fall at the top of the video – you'll want a frame where the meter stick is in clear view as well (you need to scale the image).

6. Create a coordinate system and place the origin on the video frame. To do this click on the coordinate button and you should see pink axis appear on the frame of your video. Click on the origin and drag it to the position of the ball at the top of your video (you can even rotate the axis, if you didn't align the camera properly when taking the video). Now the software will know where the origin is, click again to turn this off.

6. Click on the button and select “New→Calibration Stick”. A blue vector should appear on you video frame.

7. Click and drag the ends of this vector to align it with your meter stick. Click on the blue number next to the vector and type in how many meters this distance is equal to (by reading the markings on the meter stick). Click again to turn

this off.

8. Click on Create→Point Mass. The video sometimes goes back to the beginning so you might have to move it forward again. You can manually track the position of this mass (the ball), telling the software its location, by pressing shift and clicking in its location.

Press Shift and click on the ball The video will put a symbol on the image and the number of the frame next to it. The video will also automatically move to the next frame. Press Shift and click on the ball in its new position. Continue until the ball has fallen from view (ignore it if it bounces back up again).

Doing this manually isn't too difficult here because the ball falls quite quickly and there aren't many frames, but this software also does autotracking which is pretty cool.

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9. A table of time, x-coordinate and y-coordinate should be present in the lower right-hand corner of the tracker window. Write down these numbers. You should put these numbers in to a spreadsheet for the following calculations. For your plot, the x-axis will be the time squared (subtract the time from the first point if its not equal to zero and multiply it with itself) and the y-axis will be height fallen (multiply these by -1 to make them positive). One you have the numbers you are ready to plot the data in Octave.

10. Use Octave to plot the distance fallen (y-coordinate) as a function of time squared (t2). Obtain the slope from this graph, which should be equal to ½ g. The following script may be useful in Octave

y = [….] t2 = [….] p = polyfit(y, t2,1); plot(y,polyval(p,y), 'r'); hold on; plot(y, t2,'r*'); xlabel('Height (m)', 'FontSize', 16) ylabel('t^2 (s^2)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Typing p will give you the slope. You have to quote this value!

Repeat the above steps with your second video, for the ball with a different mass. Again, the slope is the most important measurement.

5. Grade

Obtaining ALL of the above data will earn you 50% towards your total grade for the lab. 25% each will be given for fitting a straight line to your different data and obtaining the gravitational constant. The write-up for this lab should include the title of the experiment, the names of your group members (no more than 4), the two Octave plots and a sentence stating the two measured values for the gravitational constant.

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Projectile Motion

1. Introduction

Projectile motion is a common example of motion in two dimensions. Neglecting air resistance the situation may be treated as motion with constant acceleration. A projectile is predicted to move in a parabolic path.

In this experiment you will study projectile motion and verify the parabolic trajectory by rolling a car off a pile of books (Thelma and Louise-esque). The car goes airborne and thereafter undergoes projectile motion. Let us take t = 0 to be the time when the car leaves the pile of books. At that instant, its velocity is entirely in the x-direction. The horizontal displacement is given by

If we take the end of the ramp to be at y = 0, the vertical displacement in the same time t is given by

where g = 9.8 ms-2 is the acceleration due to gravity. Eliminating the time of flight from these two equations gives the vertical coordinate as a function of the horizontal coordinate

Note that this equation is of the form

2. Apparatus

A toy car Camera connected to a computer (with “guvcview” and “tracker” installed) Meter stick

3. Procedure

1. A uniform background is required and, therefore, the experiment should be conducted near the wall at the end of the bench.

2. The software you will be using requires you to measure distance. Therefore, a meter stick should be placed against the wall and held in place to give your analysis scale.

Under “Image Controls” the Exposure, Auto should be set to “Manual Mode” Under “Image Controls” the Exposure (absolute) should be set to around 20 (any less and it looks dark). Under “Videos & Files” the frame rate should be “30/1 fps” Under “Videos & Files” you can also change where the computer will save the videos (Desktop). Feel free to play around with brightness and contrast, but note that you have to be able to read the markings on the meter stick to give the scene scale.

4. You will be investigating the projectile motion of a toy car. Place several books to one side of where the car is going to fall. Drive the car off the pile of books such that it falls down, but continues to move horizontally in the scene.

5. Press record on the software “guvcview” and roll the car off of the pile of books. Press stop to finish taking the video. You should record 3 different videos of the same car falling to make sure you have at least one good video to analyze.

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4. Analysis

You should have 3 video files but you'll only need one. You will analyze the projectile motion of the car using a software called “tracker”. The software is on the laptop provided as well as available to download:

http://www.cabrillo.edu/~dbrown/tracker/

To analyze your videos, follow the procedures below

1. “Tracker” doesn't seem to like the avi file format and prefers mov format instead. Therefore, we'll convert your chosen video file first. This can be achieved using “vlc” or using the command line:

Here it is assumed that the videos are in the Desktop folder From the main menu open up a terminal: Accesories→LXTerminal then type

cd Desktop avconv -i “capture-1.avi” -r 30 “output1.mov”

where the “capture-1.avi” is the name of your file and “output1.mov” is the name of the file your creating.

2. Open the “Tracker” software. It can be found under “Education” in the main menu.

3. Click on “Video→Import” and then go to the video (should be in mov format) you want to analyze.

4. You can step through the video, frame by frame, by pressing the button.

5. Go to the frame just as the car is leaving the ramp – you'll want a frame where the meter stick is in clear view as well (you need to scale the image).

6. Create a coordinate system and place the origin on the video frame. To do this click on the coordinate button and you should see pink axis appear on the frame of your video. Click on the origin and drag it to the position of the car (for example, select one of the wheels). Now the software will know where the origin is, click again to turn this off.

6. Click on the button and select “New→Calibration Stick”. A blue vector should appear on you video frame.

this off.

8. Click on Create→Point Mass. The video sometimes goes back to the beginning so you might have to move it forward again. You can manually track the position of this mass (the car), telling the software its location, by pressing shift and clicking in its location. It might be easier to track a particular feature of the car, such as one of its wheels.

Press Shift and click on the car The video will put a symbol on the image and the number of the frame next to it. The video will also automatically move to the next frame. Press Shift and click on the car in its new position. Continue until the car has fallen from view (ignore it if it hits the table).

Doing this manually isn't too difficult here because the car falls quite quickly and there aren't many frames, but this software also does autotracking which is pretty cool.

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9. A table of time, x-coordinate and y-coordinate should be present in the lower right-hand corner of the tracker window. You should write these numbers down and type them into a spreadsheet. For the analysis to work the numbers must all be positive and nonzero. Multiply your y-coordinates by -1 to ensure they are all positive. Add a small number to all the y-coordinates if the top number is zero or negative. For the x-coordinates, again ensure that your data is positive and nonzero. The numbers should look parabolic. In other words, the x-coordinate data should increase linearly from almost but not quite zero. The y-coordinate data should increase as x squared from almost but not quite zero.

10. Assuming that where a and b are both positive, but otherwise unknown, constant numbers. Multiplying both sides of this equation by negative one, and then taking the natural logarithm of both sides yields: . This is an equation for a straight line when is plotted as a function of . The slope of this line is b (which should be 2, right?). Using your data, plot as a function of and determine the slope to obtain b. The following script should work. You will need to include positive nonzero numbers for your x- and y-coordinates.

x=[ … ] y=[ … ] lx = log(x); ly = log(y); p = polyfit(lx,ly,1); plot(lx,polyval(p,lx), 'r'); hold on; plot(lx,ly,'k*'); legend ('numerical fit', 'Data', 'location', 'southeast'); xlabel('log(x)', 'FontSize', 16) ylabel('log(y)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Type p to get the slope and intercept. Look up (i.e., google) these commands, or come and ask me if you don't understand the script.

Sometimes the first data points are at the origin and when you put this into the logarithmic plot they can give you rogue results (points that lie outside of the obvious trend of the remaining results). If this is the case, and your data is in a straight line with the exception of one or two points then it is perfectly acceptable to remove a couple of points. Just state this in your write up.

5. Grade

Obtaining ALL of the above data will earn you 50% towards your total grade for the lab. 50% will be given for fitting a straight line to your data and obtaining the slope. Note that the slope is very important! I want to know what it is. No really I do.

Your write up should consist of the title of the experiment, the names of your group members, the Octave generated plot and your value for the slope of the straight line fit. Was the value equal to 2?

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Fractal Dimensions in Nature

1. Objective

This lab studies the fractal properties of natural patterns. The data you obtain from your measurements will be used to estimate the fractal dimensions of a rough surface.

2. Apparatus

• fractured surface (torn piece of paper towel) or a fractal object from nature (e.g., fern leaf) • graph paper

3. Procedure

You will be using a “box counting method” to determine the Minkowski–Bouligand dimension. If you 'cover' a line (in your case it will be a fracture surface or the outside of a leaf) with a series of boxes, then how many boxes will it take to cover the line? The rule is that the boxes have to be on a grid. As you can see from the example below, the number of boxes required depends on the size of the boxes. It will also depend on the roughness of the surface.

1. Trace out the outline of either a fractured surface or the outline of an object from nature (e.g., leaf) onto graph paper.

2. How many squares are covered by the outline with the smallest sized squares? This is N(1 mm) the number of squares covered for a grid of squares whose length is 1 mm. Note that your only interested in the number of squares that cover the outline and not the squares inside the outline.

3. Now consider the grid to consist of a network of squares which are 2 mm by 2 mm in size. How many 2 mm by 2 mm squares are covered by the outline? This is N(2 mm) the number of squares covered for a grid of squares whose length is 2 mm.

4. Repeat the above measurements for squares which are 5 mm by 5 mm and 1 cm by 1 cm in size.

4. Analysis

You should have 4 values (for the 4 different box sizes). Lets think about how N should vary with size. If the surface is a straight line then we might expect (It's instead of r because as the squares get smaller, more will be needed to

cover the object.) If the object is 2-dimensional, such as an entirely filled in square, we might expect .

However, for more complicated outlines, such as those considered here, we expect where b is a number between 1 and 2. For rougher surfaces we might expect a higher value for b.

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How rough is the surface you investigated?

1. First take the log of the above equation to obtain . This is of the form y = mx.

2. Plot as a function of to obtain the slope, b. 3. The slope is calculated from rise over run so

Therefore, the error in the slope, s, is given by

So given (the estimated error in N) we can estimate the error in the slope. The values of N and r in the above equation should be conservatively chosen to maximize the estimation of the error. In other words, calculate this quantity for each N and r, and pick the largest error.

Here's the Octave commands which you can use to calculate the fractal dimensions and plot the necessary graph. First enter the data for which should be calculated first using a spreadsheet or your calculator.

loginvr = [….....]

and the data for which, again, should be calculated first using a spreadsheet or your calculator.

logN = [….....]

Now its time to fit the data to a polynomial of order 1 (a straight line) and plot the results.

p = polyfit(loginvr,logN,1); plot(loginvr,polyval(p,loginvr), 'r'); hold on; plot(loginvr,logN,'r*'); xlabel('log(1/r)', 'FontSize', 16) ylabel('log(N)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Again to obtain the slope for the data entered

5. Grade

50% of your grade will be rewarded for obtaining your data. 50% will be rewarded for the error analysis (obtaining reasonable uncertainties in your measurements for the number of boxes counted), plotting versus using Octave and obtaining the fractal dimensions (slope) of the surface analyzed, along with the error in fractal dimension.

Your report should consist of the title of the experiment, the names of the group members (no more than 4), the above plot and a sentence stating the fractal dimension (with estimated error).

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Conservation of Mechanical Energy 1. Purpose

In this experiment you will investigate if energy is conserved as potential energy is converted into kinetic energy.

2. Equipment

• Meter stick • Pulley (blue adjustable)

• Vernier Caliper • Mass hanger

• Triple-beam Balance • Masses

• Air track • Photogate

3. Background

Energy conservation is one of the greatest principles in all of physics. Use of energy conservation is extremely powerful in solving many problems. Energy can take many forms: kinetic, potential, electrical, magnetic, mass, thermal, acoustic, light, etc. Today we will study mechanical energy in the absence of friction where gravitational potential energy is converted into kinetic energy. Where kinetic energy is the “energy in motion” and given by formula

Potential energy is the energy of position in contrast to the energy of motion (kinetic energy). If a mass is at a height h then the force of gravity has the potential to do work

as the mass falls. The gravitational potential energy represents the work that gravity performs as it pulls the mass down to a lower state. It also can be thought of as the potential for doing work. The main objective of this lab is to test the law of conservation of energy. We will observe how potential energy is directly converted to kinetic energy.

4. Procedure

A glider of mass M slides on a frictionless air-track. Another mass m is suspended from a massless string which passes over a frictionless pulley, and is attached to the glider M. A photogate sensor is placed above the airtrack so that the glider interrupts the sensor and the time of the interruption of the sensor is clocked. For each trial (set of m and M) the glider will be set, released, and timed three times allowing for a calculation of the error in the time, t (i.e., standard deviation). Be careful when releasing the gliders, unless they are gently caught by a student at the end of the run, they will crash into the end of the track and cause damage

During the time that the photogate is interrupted the glider advances a distance of L, the width of the glider. Knowing the time interval t and the distance traveled (width of the glider) L, we find the average velocity v by L/t. The hanging mass m receives the gravitational force mg. The whole system is driven by this force of mg. The total mass of the system is m+M, and thus the entire system moves at an acceleration given by

For a system moving at an acceleration a its velocity changes over a distance of x by

Where vi is the initial velocity and vf is the final velocity.

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Since the system has zero velocity at the beginning of the motion, we find the velocity at the distance x is given by

The experiment is aimed at a comparison of the theoretically calculated values and the experimentally measured values. The distance between the initial release and the photogate is fixed, and remains unchanged for the entire experiment. The mass of the glider M can be determined using a balance.

For the hanging mass, choose 6 different masses between 50g and 100g. With these 6 different masses m and a fixed value of M, make the measurement of the average velocity v at a fixed distance of x. Make a table tabulating the hanging mass m and the measured velocity v at the distance x, a square of the measured velocity v2, and the value of acceleration computed from the measured v and x. In the same table show the theoretically calculated value of acceleration using g = 9.8 m/s2.

Notice that, depending on the way that you set up the experiment, there may be a discrepancy between the average velocity of the glider and the final velocity of the glider. (Think about it!) Can you eliminate this error?

5. Theoretical Consideration

In the motion of these two masses, frictional force is assumed to be negligible. Thus the mechanical energy of the system is conserved. At the beginning both masses, m and M, are at rest, meaning that they have no kinetic energy. After they move the distance x, their kinetic energy increases as their potential energy decreases by mgx. Thus we can write

Notice from the equations above that

Again, this equation is in the form y = mx +c, where c = 0.

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6. Analysis

Make a graph using Octave of along the vertical axis and along the horizontal axis. Connect these data points by a straight line, and obtain its slope. Compare the slope, and your error in the slope, with the gravitational acceleration g. The octave commands for generating the plot are simply the ones required for a straight line fit

x = [ … ] y = [ … ]

where the above data is the x and y data points, and , respectively

p = polyfit(x,y,1); plot(x,polyval(p,x)); hold on; plot(x,y,'k*'); xlabel('m/(M+m)', 'FontSize', 16) ylabel('v_f^2/2x (m s^{-1})', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Again type p to obtain the slope and intercept from your curve fit. Is the intercept equal to zero? Is the slope g?

You can either add this to your graph using the command label, or you can include in your report with the graph embedded in the document.

7. Grade

50% of your grade will rewarded for collecting the above data (at least 3 different runs for each of the 6 different hanging masses). 25% of your grade will be rewarded for the error analysis of your data (errors in masses and velocities) and obtaining the error in your measurement of g. 25% of the grade for this weeks laboratory will be rewarded for your graph of

as a function of . From this graph you must obtain g, and estimate its uncertainty from the lowest values on your y and x axis, and the uncertainty in these quantities.

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Simple Pendulum 1. Objective

This lab studies the behavior of a classical periodic motion, the simple pendulum.

2. Introduction

The simple pendulum exhibits periodic motion. When the angle of oscillation is small, the motion is “simple harmonic”. In this experiment you will use simple harmonic motion, in the form of a simple pendulum to determine g the acceleration due to gravity. In the process you will also demonstrate that the period of oscillation is independent of the mass of the pendulum bob.

Consider the figure above which depicts a bob of mass, m, suspended by a string of length L from a fixed upper end. The string makes an angle with respect the gravitational force mg. The force exerted by the string on the bob is labeled by T. The tangential component of the gravitational force always acts opposite the displacement. That is, it is a restoring force. According to Newton's Second Law the sum of the forces on the bob in the tangential s direction are equal to the mass of the bob times its acceleration in the s direction. That is

assuming that the friction drag force is very small. The tangential distance (arc length) s is related to and L via . Further, if one considers only small oscillations then, , and

This is the equation for simple harmonic motion with angular frequency . The time required to complete one cycle of motion is the period T, and is given by

This expression predicts two things. First, the period is independent of the bob mass m. Second, a plot of T2 as a function of L should be a straight line with slope .

You will measure the period of motion of the simple pendulum for several different string lengths using several bobs of different weight. You will then plot the period squared versus the string length for all bobs, and determine g from the slope.

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3. Apparatus

• pendulum clamp • protractor • bob (hanging weight) • clamp stand • string • stop watch • meter stick

4. Procedure

In taking your data keep the angular displacement below 20°. Check this with a protractor if needed. At larger angles the motion is no longer simple harmonic! When selecting the string lengths L do not go to the trouble to have exact numbers (like 0.50 m). Just hang it approximately, and then measure the length. Finally, make sure that you do not use identical (or nearly so) values of the string length for any of the two hanging masses. Doing so will enable all your data points to be clearly visible on one plot.

• Make a table in your notes with columns for the length L, mass m, the average period T, and T2.

• Select one of the bobs (hanging weights) and mount it on the stand with length approximately 0.5 m.

• Measure L the distance between the top of the string and the center of the bob. Do this by measuring from the top of the string to the top of the bob and then adding the radius of the bob. Note , the error associated with this measurement.

• Start the pendulum swinging with small oscillations to ensure simple harmonic motion. Use the stop watch to measure the total time for 10 complete periods (motions back and forth). The result divided by 10 is then the period. Repeat this method of measuring the period nine times. Calculate the average and standard deviation of these nine measurements. This is at this length, for this bob.

• Repeat the above procedure for five more lengths between 0.50 m and 0.30 m. Then repeat for the other bob (different weight). You should have measured the period at six different lengths for each of the bobs when you are finished. Do not use exactly the same lengths for your measurements. Scatter the lengths so that your data points will not be exactly on top of one another when you graph the data.

• Using your measurements for the period T, calculate T2. may be obtained using .

Just to clarify, you will have 108 measurements of time. 9 measurements (to get good statistics with your average and standard deviation), for six different lengths, and for each bob (mass). Once you have all 108 measurements, put this into a spreadsheet and calculate the averages and standard deviations. You should have 12 averages and standard deviations for the time, which you can use to calculate 12 measurements of T2 and , as described above.

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5. Analysis

Make a plot of T2 versus L in Octave. Label the data from each bob using different symbols. Calculate the slope b, for the sum of all two data sets. Calculate the uncertainty (ask if you don't know how, or recalling slope is simply rise over run look at the equations for calculating errors from the first lab).

Determine the acceleration due to gravity from the slope b using

and

The Octave commands required are as follows:

x1 = [ … ] y1 = [ … ] x2 = [ … ] y2 = [ … ]

where the above data is the x and y data points, L and T2, respectively, for both bobs.

The following commands can now be used to both plot the necessary figure and obtain the slope

p1 = polyfit(x1,y1,1); plot(x1,polyval(p1,x1)); hold on; p2 = polyfit(x2,y2,1); plot(x2,polyval(p2,x2)); plot(x1,y1,'b*'); plot(x2,y2,'r*'); xlabel('L', 'FontSize', 16) ylabel('T^2', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

You should add a legend to this script as well. The slopes can be obtained by typing p1 and p2.

6. Grade

50% of your grade will be rewarded for acquiring the measurements described above. 25% will be rewarded for calculating the correct uncertainties. The remaining 25% will be rewarded for your plot of T2 versus L, and the determination of two separate g's. The plot must be completed in Octave in the usual way.

Your lab report should consist of the name of the experiment, the names of the group members (no more than 4), a plot clearly showing the different lines for the different bobs, and the two different measurements of g along with errors.

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Atwood’s Machine

1. Introduction

Newton’s First and Second Laws describe the relationship between force, mass, and acceleration. The net (total) force on an object, which is the vector sum of all applied forces, is equal to the total mass times the acceleration of the object. In this lab you will use a simple system of pulleys, strings and weights to investigate this relationship.

2. Equipment

• low friction pulley & clamp • string • 2 mass hangers

• 2 sets of masses • stopwatch • meter stick

3. Theory When two masses are suspended by a string over a pulley each feels a downward force due to its weight (W=mg) and an upward force due to the tension (T) in the string. If these two forces are equal, then the net force on the mass is zero and, as Newton’s first law tells us, there will be no acceleration of the mass. In an Atwood's Machine, the difference in weight between two hanging masses determines the net force acting on the system of both masses. If one mass is heavier than the other, then the masses may accelerate – one moving upward and the other down.

If the string and pulley are considered to have negligible mass, then we can neglect the force needed to accelerate the string and cause the pulley to rotate. In this case the tension in the string is equal on both ends.

We will use a for the acceleration of the lighter mass and -a for the acceleration of the heavier mass (ignoring the fact that acceleration is a vector). Using this substitution we can write the following equations

where we have used the fact that the acceleration of each mass is the same but in opposite directions (since they are connected by the string) and assumed that mass 1 is accelerating upward and mass 2 is accelerating downward. Subtract one equation from the other to cancel out the tension

Algebraically we can solve this to show that

Thus, we can see that if m2 is greater than m1 (as assumed), m2 will indeed accelerate downward.

The acceleration may be determined by measuring the time, t, required for the mass to fall from rest a specific distance h. If a is constant then we know that

Note: masses crashing into the floor is generally a bad thing. Please take precautions.

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4. Procedure

You will be investigating the relationship between mass and acceleration by varying both m1 and m2 and observing a. We have shown from Newton’s Second Law that

The above equation is in the form y=mx + b where b (the intercept) is 0 and, therefore, a graph of a (acceleration) versus will give you a straight line with a slope of g.

1. Raise the heavier mass and hold it steady. Wait until any swinging motion stops. In order to avoid very complicated motions you should attempt to minimize “swinging” or other motion as the mass descends.

2. Release the heavier mass and determine the length of time required for the mass to fall and the distance it falls.

3. Repeat this for a total of seven trials to find the acceleration a and its uncertainty .

The error in t2 can be obtained first using , and then you can use the equation for propagating errors in equations with division to find the total error in a, .

4. Change the masses and repeat your measurements of the acceleration at least ten times.

5. Using Octave make graphs of a versus for your data set. Find the slope of your graph, the gravitational acceleration, g, and the uncertainty in your value using

Octave commands are given below:

a = [ … ] m = [ … ] p = polyfit(m,a,1); plot(m,polyval(p,m)); hold on; plot(m,a,'k*'); xlabel('m/(M+m)', 'FontSize', 16) ylabel('a (m s^{-2})', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

5. Grade

50% of your grade will be rewarded for performing the experiment and obtaining all of the data specified in the procedure. Correct error analysis will account for 25% of your grade for this laboratory and the remaining 25% will be given for your Octave graph and the determination of g. The error in g can be obtained from your error in acceleration .

Your report for this experiment should consist of the title of the experiment, the names of your group members, the Octave plot and the measured value of g (along with the error).

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Hooke’s Law and Simple Harmonic Motion

1. Introduction

In this experiment you will determine the force constant, k, of a given spring by application of Hooke's law. Furthermore, you will show experimentally that the period, T, of the motion of a mass, m, hanging from a spring depends on m and k.

2. Equipment • stop watch • set of slotted weights • spring

• meter stick • rod and support

3. Background

A periodic motion is one that repeats itself in successive equal intervals of time, the time required for one complete repetition of the motion being called its period. If a mass is to be caused to execute simple harmonic motion, the force acting on it must be proportional to the displacement of the mass from its center or equilibrium position and directed toward that position.

Suppose, for instance, that the mass is suspended from a spring. If, when the mass is above or below its equilibrium position, there exists a restoring force proportional to its displacement from equilibrium then simple harmonic motion occurs. For a Hooke's law restoring force, the relationship between the force and the displacement is given by F = -kx where k is called the “spring constant” and x is the displacement from equilibrium. Application of such a force to a mass m yields

which is the mathematical statement of the condition for simple harmonic motion. When x = 0 the mass is at the center or equilibrium position. Note that the restoring force must exist for both positive and negative x. The resultant equation of motion of the mass (variation of x with time) is given by

where is the period, A is the amplitude of the motion, and t is the time elapsed starting from x = 0. Knowing that acceleration is the second derivative of displacement, can you prove the above expression satisfies the condition for simple harmonic motion?

The following procedure is split up into two parts. The first measures the spring constant (force divided by distance stretched) by hanging weights, and the second measures the spring constant indirectly through the period of oscillation. However, you can do the two parts simultaneously using the same weights if you want. Furthermore, to ensure comparison, make sure all of your numbers are kept in SI units.

4a. Procedure for Part 1

1. Place a 50g mass hanger on the spring. Record the equilibrium position of the hanger.

2. Apply a series of additional forces to the spring by hanging various weights.

3. Using Octave plot force added to the hanger versus displacement of the hanger from equilibrium, and determine the force constant k of the spring from your graph. You can find k by determining the slope of the best fit line; make sure that you include this plot and the resulting line fit when you hand in your data.

4. Find k.

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The Octave commands for generating the above plot are given below:

f = [ … ] x = [ … ] p = polyfit(x, f,1); plot(x,polyval(p,x)); hold on; plot(x,f,'k*'); xlabel('x (m)', 'FontSize', 16) ylabel('F (N)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16) 4b. Procedure for Part 2

1. Determine the period of oscillation for at least 5 different masses added to your spring. Time at least 20 oscillations for each load. Divide the time required for 20 oscillations by 20 to determine T. It would be prudent to repeat the measurement for each mass at least 3 times (or have 3 different team members independently time the oscillations) so that you can find a T.

2. Note that since , we can write for a simple harmonic oscillator. Plot T2 versus m to obtain k, and the uncertainty in k. Compare this value of k with the value obtained in the first part of your experiment (using Hooke's law).

T2= [ … ] m = [ … ] p = polyfit(m T2,1); plot(m,polyval(p,m)); hold on; plot(m, T2,'k*'); xlabel('Mass (kg)', 'FontSize', 16) ylabel('T^2 (s^2)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Typing p gives you the slope, , which can be used to calculate k, the spring constant.

To find the error, first find the error in T2, The equation on the bottom of page 10 tells you how to calculate the error in a division. Use this to find the error in the slope. If b is the slope and is the error in the slope then error in the spring constant is

5. Grade

There are two parts to this laboratory, with each part worth a total of 50% towards the total grade. 50% will be rewarded for obtaining the data , 25% will be rewarded for including the correct error analysis, and 25% for obtaining two values of k from the Octave plots.

Your report should contain the title of the experiment, the names of your group members (no more than 4), the two Octave plots and your two measured values of k along with the uncertainties.

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Sound Waves

1. Introduction

Sound waves exist as variations of pressure in the air. What we perceive as pitch is the frequency of the wave. However, most sounds are not a single frequency or tone, but a superposition of many frequencies. In this lab you will record some sounds and consider the frequencies in the recorded sounds.

2. Equipment

A computer with “Audacity” installed. A USB microphone. Some musical instruments to analyze.

3. Background

We will be using Fourier Transforms to investigate the sound waves. The Fourier Transform essentially takes a time series of pressure intensity (that is what the sound wave is) and determines which perfectly pitched frequencies would have to be added together, and also in what amplitudes, to recreate the original sound. In other words, you go from a series of data which tells you the amplitude of your sound as a function of time, to a new series of data which tells you the amplitude of the frequencies that would have to be added together to recreate the sound. Usually, we would say that we have gone from the time-domain to the frequency-domain. This can be useful for a number of reasons. For example, say you wanted to remove all high-pitch noise from a signal. Simply perform the Fourier Transform of your original signal to enter the frequency-domain, put the amplitudes of the high frequencies to zero and then perform a back Fourier transform to return to the time domain. Now you have the original signal, minus any high frequency noise.

This isn't limited to sound waves, however. In mammography, the scanned data is subject to such Fourier Transform filtering to isolate critically sized clusters of microcalcifications and characteristic masses. Such computer aided diagnosis (CAD) can find tumors missed by a physician; statistics from a recent study suggests that a diagnosis with a single physician and a CAD combined is almost identical to a diagnosis with two physicians. Here, however, we'll take the Fourier Transform of different sounds to analyze the frequencies that make up these sounds.

4. Procedure

1. Open up Audacity. It is under Sound & Video from the main menu.

2. The first time you record something go to “Edit→Preferences”. Click on devices and select the recording device (which should give you a drop down menu). Select the USB microphone

AK5370: USB AUDIO

3. Press the record button and record your sample

4. To obtain the Fourier Transform go to Analyze→Plot Spectrum. This will open up a new window.

5. Click on size and select the highest value possible (16384).

6. Click on export and export the data to a text file (you can change the name from spectrum.txt). This text file has two columns “Frequency (Hz)” and “Level (dB)” which can be opened and manipulated in a spreadsheet.

8. Record a tuning fork of known frequency for calibration, and one other sound that you would like to analyze. Obtain the Fourier Transform of these two signals.

9. You, therefore, should have 2 text files with the Fourier Transform of the tuning fork and one other sound.

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5. Analysis

Open up the text files with the Fourier Transform of the signals in a spreadsheet. You only really need the data from around 200 Hz to about 1000 Hz, so you can isolate this region of the spreadsheet (about 300 data points).

Plot the 2 signals on the same plot using Octave. Importing data from Excel to Octave should be done by first copy and pasting this script into notepad (or other minimalist text editor with no formating), then pasting the numbers into the script from Excel, and then copying the commands and numbers from notepad into Octave. This way you'll have a permanent record of the commands used to generate the plot, in case it messes up and you need to redo anything.

A possible script could be

f = […..] s1 = [….] s2 = [….]

plot(f, s1,'k-'); hold on; plot(f, s2,'r-'); hold on;

xlabel('f (Hz)', 'FontSize', 16) ylabel('L (dB)', 'FontSize', 16) legend ('Tuning Fork', 'Drum', '', 'location', 'southeast'); axis('tight') set(gca,'FontSize',16)

Note your legend will reflect the sound wave you choose to analyze.

6. Grade

Performing the experiment and handing something in is worth 50% of the grade. The remaining 50% is for producing a graph comparing the Fourier Transforms of the 2 different signals (appropriately labeled) and identifying within your data the frequency of the tuning fork.

Your report should consist of the title of the experiment, the names of your group members, and the Octave plot.

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Greenhouse Effect

1. Introduction

Given the potential ecological and social problems that we may face as a consequence of climate change, it may be important for you to understand the physics behind the greenhouse effect. In this experiment a higher temperature is expected to be observed in a container with elevated concentrations of CO2 inside, than in a container with just air enclosed, when subject to direct light. The CO2 absorbs outgoing thermal radiation and causes the air inside the container to be warmer. This is the greenhouse effect.

2. Equipment

• thermometer • polypropylene container with black lid • balloons • plastic bottle • vinegar and baking soda.

3. Procedure

1. Half of the class will investigate containers filled with air, while the other half of the class will investigate containers with elevated concentrations of CO2 inside. Therefore, the groups will be divided equally by the instructor.

For groups looking at containers of air

2. If you are investigating the container with just air inside then all you have to do is place your container upside down in the sun. Make sure the container is adequately sealed, that the temperature probe from the thermometer is inserted into the hole of the container and measure the temperature every 10 seconds for 10 minutes (or until an equilibrium temperature is obtained).

For groups looking at containers with elevated CO2

3. If you are investigating a container with elevated CO2 concentrations inside then you must first create the CO2. Mix vinegar and baking soda in a plastic bottle and quickly place a balloon over the top of the bottle. As CO 2 is produced by the reaction, the balloon will fill with CO2.

4. Pinch the balloon, to stop the CO2 escaping and remove from the plastic bottle. You will release the CO2 into the polypropylene container and quickly close the lid, sealing some of the CO2 inside.

5. Place your container upside down in the sun. Make sure the container is adequately sealed, that the temperature probe from the thermometer is inserted into the hole of the container and measure the temperature every 10 seconds for 10 minutes (or until an equilibrium temperature is obtained).

6. Once you have finished and you have all of your data, ensure that other groups have your final temperature and collect the final temperatures from other groups. Presumably, the temperature inside the container will have settled to an equilibrium after 10 minutes, but it can sometimes take longer.

7. You will plot your data of temperature as a function of time and include a table of final temperatures for all of the experiments conducted within the class. The table should have two columns, with temperatures from the experiments containing just air in the left-hand column and temperatures from the containers with elevated CO 2 concentrations inside in the right-hand column.

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4. Analysis

The plot of temperature as a function of time for your experiment should be plotted using Octave and the following script might help (it is only a simple x-y plot):

t = […..] T = [….]

plot(t, T,'k-');

xlabel('Time (s)', 'FontSize', 16) ylabel('Temperature (^oC)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

Create a table of final temperatures for all of the experiments conducted within the class. The table should have two columns, with temperatures from the experiments containing just air in the left-hand column and temperatures from the containers with elevated CO2 concentrations inside in the right-hand column. Calculate the average and standard deviations for the two columns. Did your class demonstrate the greenhouse effect? While this experiment does not replicate the Earth (we don't live in a plastic container for one) it does demonstrate how raising CO2 concentrations can result in warming.

5. Grade

50% of the grade will be performing the experiment, 25% will be for including your Octave plot of temperature as a function of time and the final 25% will be for the table showing the equilibrium temperatures obtained from all of the groups in class.

Your report should consist of a title for the experiment, the names of your group members, the Octave plot and the table of equilibrium temperatures (including the average and standard deviation).

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Bernoulli's Principle

1. Introduction

In this weeks experiment we will explore Bernoulli's equation. The equation can be written as

where P is pressure, is the density of the fluid, g is the gravitational constant, h is height and v is velocity. Bernoulli's principle states that the above expression is always constant. Therefore, say we reduce the pressure term. The total expression has to remain constant, therefore, the velocity term must increase to compensate for the decrease in pressure.

In particular, if the fluid is moving horizontally, then (contrary to what you might guess) the pressure P will be lower where the fluid is moving faster. This certainly surprised many of Bernoulli's contemporaries, and it still surprises people today. But it explains, for instance, how squirting works. You fill a balloon with water and squeeze it. The pressure P is high inside, and low outside. The water is moving slowly inside, and fast outside. The equation really does make sense!

The equation applies only to fluids in steady flow along a single path followed by a particle of fluid. In steady flow, such a path is called a streamline. At any point on the streamline, you can add up the three quantities on the left of the Bernoulli equation. The equation says that if you do that for any two points on the same streamline, then the two answers will be the same.

2. Method

Take an empty plastic cylinder with a hole near the bottom. Put a finger over the hole and fill the container with water. When you take your finger off the hole, the water will emerge in a jet.

We can define h to be the vertical height of the water above the hole. Once you take your finger off the hole the pressure at the jet and the top surface become approximately equal to atmospheric pressure so we can treat P as a constant. We assume that the water is incompressible and so treat as a constant too. Recall that the left hand side of the Bernoulli equation adds up to a quantity that is constant all the way along the streamline. So, we can write the equation for points on this streamline

where k is a constant which now includes the pressure/density term of the original equation. If the surface of the water is at height h = H then, assuming that the velocity at the surface is very small, and it can be set to zero for the purposes of this calculation, then . The jet of water is at height h = 0 so its velocity, v is approximately given by

In other words, the equation predicts the velocity v of the jet to be equal to

In practice, measuring the velocity of the jet is difficult so we'll measure it indirectly by tabulating the height of the water in our cylinder, H, against the elapsed time t. Observe that the rate at which H changes is proportional to the speed at which water is leaving the cylinder: in other words, the velocity of the jet.

Therefore, you need to measure the height of the water above the hole in meters as a function of time in seconds. We'll repeat the experiment twice to get better statistics.

Phys 1215 Laboratory Manual Spring 2014 37

3. Analysis

You should obtain a graph which looks similar to the following

Note that the curve can be fitted to the parabolic curve, or second order polynomial.

If the rate of change in height is proportional to velocity then

and placing this into our Bernoulli equation gives

Of course we know that H is decreasing, so its derivative is negative

rearranging and integrating gives and the constant C can be obtained by taking the height at t = 0 to be H0. The final equation is of the form

From a polynomial fit to your date (height versus time) obtain a numerical value for . In other words, match the parameters of your fit with the expressions above and solve for (you should obtain two values) The Octave commands to do this are over leaf. Ask if you need help.

H = [ ... ] t = [ … ] p = polyfit(t, h,2); plot(t,polyval(p,t)); hold on; plot(t, h,'k*'); xlabel('t (s)', 'FontSize', 16) ylabel('H (m)', 'FontSize', 16) axis('tight') set(gca,'FontSize',16)

typing p will give you three numbers corresponding to the variables a, b and c in the equation

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Comparing the curve fitted parameters with the above equation we can obtain

Now we know what H0 is (its just the initial height at t = 0) so we can see if the fit is OK by checking the value of c which should be reasonably close. Using the equations for a and b, we can obtain two values for . Note that because you perform the experiment 3 times, overall you will have 6 values of , which should all be reasonably close.

Note that while the polyfit function generates the best fit to your data (minimizing the square of the difference between your data and the function) it doesn't tell you how good of a fit this is. This is somewhat of a mystical art (as we saw with the straight line fits where we took a conservative estimate). A plot will certainly help. The question really comes down to, how accurate would the fit be if I varied the parameters? There are statistical ways to evaluate (ahem fudge) this, but a visual estimate is good enough here. To find the error in a and b, just vary them until they look terrible!

p2 = [a2 b2 H0] plot(t,polyval(p2,t)); hold on; plot(t, h,'k*');

where a2 and b2 are the coefficients from your fit, which you can tweak (individually) until the fit is subjectively deemed to have broken down and the coefficient c was essentially H0 anyway (or it should have been) so we can just keep that value fixed. Given the “acceptable” variations in the coefficients a and b, what are the variations in ? Ask if you need help!!!!

4. Grade

Obtaining the results from your experiments is worth 50% of this lab grade. The correct curve fitting of your data, obtaining 6 different 's in the process, is worth 25% and estimating the error is worth the remaining 25%. This error can be obtained from plotting polynomials with slightly different coefficients as described above. Within your estimated errors, do the six 's agree with each other?

Your report should consist of the title of the experiment, the names of your group members, one of the Octave plots used to calculate the 's and a table containing the six values of along with the six uncertainties .