Lab Assignment
Name: _____________________________________
PHY 111: Remote Lab #1
Data Analysis
Purpose:
To investigate how measurements are performed and analyzed in physics
Objectives:
· To describe the process of measurement
· To present data in tabular form
· To properly plot data in graphical form
· To analyze data in graphical form
· To determine the standard deviation of a set of measurements
· To calculate various forms of uncertainties
Materials & Resources
· Various disks
· String
· Scissors
· Meter stick and/or metric ruler
· Graph paper
· “Accurate masses and radii of normal stars: modern results and application”, Torres, Andersen, and Gimenez, Astron. Astrophys. Rev. 18, 67-126 (2010)
Introduction
How do physicists make and report measurements?
Very often, physicists deal with “quantities”, concepts that can be expressed in numerical form. Quantities may come from direct measurement or calculation; they may refer to everyday experiences or abstract notions, they may be expressed in words or symbols, but in the end must be based on numbers.
The emphasis on quantities is not arbitrary. We may have many notions (“theories” or “models”) to use to describe how nature operates, but it is impossible to test the validity of any particular notion without comparing it to nature – which is where quantities come in.
Part #1: Determination of – calculation method
Before talking in detail about the meaning of the concept of “quantity”, let’s perform some measurements and analysis of one.
The number “pi” (or, in Greek, “”) is amongst the most important numbers in all of mathematics and physics. It is difficult to determine – for example, it is an “irrational” number, meaning it cannot be found by taking the ratio of two whole numbers. Yet that is its most simple definition – pi is the ratio of the circumference of a circle compared to its diameter. In equation form,
If we cannot simply calculate as a ratio, what can we do instead? One approach is to approximate pi through measurements.
Here, we will take a series of measurements that will allow us to calculate a value for . Along the way, we will also discuss some issues that come up while making measurements and how we should report them.
1. Before we begin, how close to do you think your measurements should be in order to be considered reliable? Within 1%? 5%? 10%? (There is no wrong answer, but we will return to this idea later)
2. Procure the following materials:
· String
· Scissors
· Meter stick
· Ruler
· 5 disks of different sizes
3. Cut or find a piece of string long enough to easily go around the largest disk.
4. Use the ruler to measure the diameter of each disk (use units of centimeters). Make sure the disks are each different in size. Record the results in Table 1-1.
5. Wrap a length of string around a disk (it doesn’t matter which one) so that the string overlaps itself. Note the positions on the string where it overlaps (see Figure 1-1).
6. Keeping these positions, straighten the string and use the meter stick to measure the length between the overlapping positions. This will be the circumference of the circle. Record the result in Table 1-1. Then repeat the measurements for the other disks.
Figure 1-1. Measuring the diameter (left) and the circumference (right) of a disk.
Table 1-1. Measurements of the Properties of Disks
Brief description Circumference Diameter Ratio
of disk (C) (D) of C to D
____________________ __________ __________ __________
____________________ __________ __________ __________
____________________ __________ __________ __________
____________________ __________ __________ __________
____________________ __________ __________ __________
Average: ___________
7. Calculate the ratio of circumference to diameter for each disk and record the results in Table 1-1.
8. Calculate the average of the ratios and record the result in Table 1-1. Show your work here:
9. The real (or “accepted”) value of is 3.1416… (Here we show 5 significant figures). To determine how close your value is to the accepted value, calculate the percentage error (show your work):
10. Do you consider your value of to be close to the accepted value? Briefly defend your answer (refer to question #1 above).
11. List some potential benefits to determining the value of in this fashion:
12. List some potential drawbacks to determining in this fashion:
13. Why do you think we made several different measurements? Wouldn’t one measurement be enough?
Part #2: Determination of – graphical method
On a separate sheet of graph paper (see the back of this document), plot the circumference as a function of diameter. To make this useful, do the following:
1. Give the graph a title. The title should express the purpose of the graph. Something like “Circumference vs. Diameter for Several Disks” would work.
2. We want to make as clear and large a picture as possible. Compare the largest circumference to the number of squares available on the vertical, in order to choose an appropriate scale. Then do the same for the diameter & the horizontal. It may be that the scales are the same, but they don’t have to be.
3. Label each axis appropriately – “circumference (cm)” on the vertical and “diameter (cm)” on the horizontal. What is the purpose of the (cm) notation?
4. Plot the points.
5. Using a ruler, draw a straight line through the points, beginning at the origin. Do NOT connect the dots; rather, draw a line (as best as you can eyeball it) that goes through the dots. Some dots may be above the line & some below, but it should not be the case that all the dots are either above or below.
6. Why did we specify starting at the origin? (This is true for this plot of circumference vs. diameter, but doesn’t have to be otherwise)
7. Pick a point on the line (not one of the dots) and measure its vertical and horizontal position, which are sometimes called “rise” and “run”, respectively. Record their values in the spaces provided below.
8. Calculate the slope of the line by dividing the rise over the run. This is another estimate of the value of :
Rise = ______________
Run = _______________ ________________
9. Is this value of the same that you determined earlier?
10. Are there advantages to determining in this fashion (especially compared to the calculation method used earlier)? Any disadvantages?
11. Which version of (calculated or graphed) gave you a closer answer to the accepted value of ? How can you tell?
12. In ancient times, the most popular approximation for was the fraction 22/7. Calculate this in decimal form and record below:
22 / 7 = ________________
13. Find the percentage error for this version of . Which estimation ended up being closer to the accepted value – your calculation, your graph, or 22/7?
% error for 22/7: __________
14. If we can have any precision we wish for (supercomputers routinely churn out millions of digits), why would anyone, now or historically, use an approximation?
Part #3: Interpretation of Graphs
Our next task is to learn how to interpret graphs – to see when they tell us something and when they don’t; to see when they tell us a relationship is simple and when it’s not.
We’ll start by plotting data from the nearest known (as of the year 2012) stars. A table of various properties of stars is provided, and we will try to learn how stars behave according to this data. Note: the symbol “ʘ” refers to the Sun – for example, 1 Mʘ represents the mass of the Sun.
1. Quickly review Table 1-4. What type of properties do you see?
2. To start, we will review potential relationships between radius and mass & between distance and mass. Before making your graphs, do you think there should be any correlation between radius and mass? Between distance and mass?
3. Using a separate sheet of graph paper, plot the radii (vertical axis) vs. mass (horizontal axis) for this set of stars.
4. Is there a relationship between radius and mass for stars? If so, what is it? If not, why not?
5. On another sheet of graph paper, plot distance vs. mass.
6. Is there a relationship between distance and mass for stars? If so, what is it? If not, why not?
Summary
Now it’s time to address questions about the quality of our results more carefully. It is likely you have seen statements in the media to this effect: “A recent poll found 75% of all college students admire their physics professor, with an uncertainty of plus or minus 3%”. Where does the 3% come from? And what does it mean?
1. Let’s make use of the calculations you made earlier for p. Copy the ratios of C/D (which we will call x, to distinguish it from ) into Table 1-5 below:
Table 1-5: Standard Deviation Calculations related to Measurements of
x x – (x – )2
______ _________ __________
______ _________ __________
______ _________ __________
______ _________ __________
______ _________ __________
2. For each value of x, subtract from it and record the result in Table 1-5.
3. Then square the result of the subtraction and record those values in Table 1-5.
4. Sum all the values in the last column (the “(x – )2” column) and record the result below:
= __________
(Note that “i” simply means each individual value of x)
5. Calculate the standard deviation of your sample of data using this formula:
= __________
6. Now re-express your answer for p (from Part #1) as your average estimate for (remember we called it “x” before) plus or minus the standard deviation:
x +/- = _________ +/- __________
How does the standard deviation help us understand our results? Suppose you repeated the measurements with five different disks. Then, roughly, you could expect the new average result to be within one standard deviation (that is, higher or lower than your original average by one standard deviation) about 2/3 of the time.
If you expand your range around the average to two standard deviations, you can expect new averages to be within that range about 95% of the time; 99.7% for a range defined by three standard deviations, and so on.
7. If you hear or read a scientist claim a “confidence” in a result, what do you think that means? How might a scientist express a “confidence level” in mathematical terms?
8. Convert your standard deviation into a percentage via the following and record below:
% standard deviation = ________ %
9. Finally, let’s return to your original thoughts regarding whether your average calculation for p is “reliable” (refer to the very first question in this lab!). How well did you do? How might you express reliability in the future?
Extra Credit
Why would we measure the value of at all? Why not plug it into one of many available math formulae instead?
A very common way to have a computer calculate is via an infinite series. An infinite series is a simply a sum, over and over again, of a set of numbers. The set of numbers in turn is determined by a formula. For example, consider the following infinite series:
Simply asks you to add reciprocals (1 divided by the number) over and over again, changing the next number by 1. Sometimes the end result keeps growing (such as the one above), which we call a diverging series; sometimes the end result settles on a particular number, in which case we call it a converging series. is a set number, so we will use a converging series to determine its value.
Let’s try a relatively simple converging series to find the value of :
Meaning, if we keep at it long enough, we can use this formula to find the value of to any precision we desire.
Let’s try it. We’ll show a few examples here, then let you try.
If n = 1, then we only have one number, 1/12, which is 1, of course. That isn’t close to , obviously, but we have to continue the calculation. To find the series value for , we need to flip around the calculation: first, multiply by 8, then take the square root:
(The “≈” sign means “approximately equal to”)
This result isn’t really close to either, so again we continue. Let’s show the next step:
If n = 2, we need to add together the first two terms in the formula, then multiply by 8, then take the square root:
1.11*8 ≈ 8.88
Note this is closer to , but still far enough off that we’d like to do better.
1. Continue this series of calculations to estimate the value of for increasing values of n. Record the results in Table 1-5:
Table 1-5. Series Calculations of
n Sum Sum*8 Estimate of
1 1/12 = 1 8.00 2.83
2 1/12 + 1/32 = 1.11 8.88 2.98
3 1/12 + 1/32 + 1/52 = ______ ______ ______
4 1/12 + 1/32 + 1/52 + 1/72 = ______ ______ ______
5 1/12 + 1/32 + 1/52 + 1/72 + 1/92
= ______ ______ ______
6 1/12 + 1/32 + 1/52 + 1/72 + 1/92 + 1/112
= ______ ______ ______
2. Are the calculations in fact getting closer to as we increase n? If so, are the results getting closer to very “quickly”?
3. A computer can calculate a reasonably precise value of quickly, but what if you had to do this by hand? (Or, was 22/7 a better option before computers?)
Table 1-4. Properties of a Sample of Stars.
Name Mass Radius Temperature Luminosity Distance
(Mʘ) (Rʘ) (K) (Lʘ) (pc)
Sun 1 1 5760 1 0
V3903 Sgr 27.27 8.09 38,000 122,000 1,390
AH Cep 16.62 7.43 30,500 42,800 687
V453 Cyg 13.05 5.64 27,800 38,300 1,780
QX Car 9.25 4.29 23,800 5,310 767
CV Vel 6.09 4.09 18,100 1,610 603
DI Her 5.17 2.68 17,000 540 630
GG Lup 4.11 2.38 14,800 241 147
IQ Per 3.50 2.45 12,300 123 271
PV Cas 2.82 2.30 10,200 51.4 641
TZ Men 2.48 2.02 10,400 42.8 118
Beta Aur 2.38 2.77 9,350 52.5 25.0
YZ Cas 2.32 2.54 10,200 62.7 99.5
V885 Cyg 2.23 3.39 8,150 45.4 836
EE Peg 2.15 2.09 8,700 22.5 139
EI Cep 1.77 2.90 6,750 15.6 228
PV Pup 1.56 1.54 6,920 4.90 84.6
DM Vir 1.45 1.76 6,500 4.45 193
V1061 Cyg 1.28 1.62 6,180 3.42 158
EW Ori 1.17 1.13 5,970 1.47 170
V1174 Ori 1.01 1.34 4,470 0.641 378
GU Boo 0.610 0.627 3,920 0.0834 133
YY Gem 0.599 0.619 3,820 0.0733 13.6
CU Cnc 0.435 0.432 3,160 0.0167 7.4
CM Dra 0.231 0.253 3,130 0.0055 6.6