Physics 174 - Exercise 4

Propagation of Errors

Purpose

The purpose of this lab is to teach you about the "propagation of errors" and how to decide experimentally if a theory is true. To do this, we have chosen a simple problem: experimentally determine if the Pythagorean Theorem is true.

The Pythagorean theorem just says that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse of the triangle; i.e. a2+b2 = c2. In this lab, you will measure the sides of some right triangles and apply error analysis to see if this law is correct.

Measuring with a Ruler

(1) Set up a spreadsheet with rows and columns as shown in the Table at the end of this write-up. Be sure to type in your name and section.

(2) Use the ruler to measure the lengths a, b, and c of the sides of both triangles. Fill in the cells in the first two empty rows of your spreadsheet.

(3) Use the spreadsheet to compute d=(a2+b2)1/2. d is the expected length of the hypotneuse if the Pythagorean theorem is true and your meaurements are accurate.

(4) Compute r=c-d for the two triangles. Note: The quantity r is the difference between theory and experiment. If there were no errors, and the Pythagorean theorem were true, we would have r=0.

Save your file

 

Estimating the Experimental Error

(5) Estimate the uncertainties in your measurements of a, b, and c. Fill in the appropriate columns in your spreadsheet.

(6) Given your uncertainties, use your spreadsheet to compute Dd, the error in d, and Dr, the error in r=c-d for both triangles. You will have to propagate the errors.

The formula for the error in d is:

[1.1]

The symbol d/b stands for partial differentiation of d with respect to b. The basic idea is quite simple. The quantity d=(a2+b2)1/2 depends on two variables, "a" and "b". d/a just means take a derivative of "d" with respect to "a" assuming that all other variables, such as "b", are constants. With a little work, it is easy to see that d/a=a/d, d/b=b/d and that Equation 1.1 can be rewritten as:

[1.2]

For this lab, Da=Db and you can show that the expression simplifies to just:

Dd = Da. [1.3]

Similarly, the formula for the error in r can be written as:

[1.4]

Working this out, we find:

[1.5]

Was Pythagoras right ?

You now have two measurements of r, the difference between theory and experiment, from two different triangles. You can use your results to experimentally test the Pythagorean Theorem. Since these measurements were taken using a ruler, they are not very precise.

(6) Use your data from Part A to compute <r>, the mean difference between theory and experiment:

Here r1 is your measurement of r for the first triangle and r2 is your measurement of r

for the second triangle.

(7) Use your spreadsheet to calculate the estimated uncertainty in r:

(8) Next, use your spreadsheet to compute 2 for your data . For this experiment, rtheory=0 if the Pythagorean Theorem is true, so you can write:

Now ri is just the difference between theory and experiment for the i-th measurement. If the theory is right, then the only difference between the theory and the experiment should be due to measurement error, so that we expect r1~r2~Dr. Thus, each term in the sum should be about equal to 1. Since there are two terms in your sum for c2, we expect c2 ~ 2. Fill in your spreadsheet and save your file.

(9) To test whether the Pythagorean Theorem is statistically consistent with the experiment, compute c2/n . This is called the "reduced value of c2" and should typically equal about 1. The degrees of freedom, n, is equal to 2 since you made two measurements and you did not use any fitting parameters. Use the table of P(c2,n) at the end of this writeup to find the probability that random errors can cause c2 to be larger than you found. Again, fill in your spreadsheet .

QUESTION #1: Is the pythagorean theorem true ? Explain briefly.

 

Measurement Uncertainty for the Calipers

(11) Estimate the uncertainty in measurements of length using the calipers.

(12) Use the calipers to measure the thickness of the larger triangle.

(13) Compare your result with the rest of the class by constructing a Table of the classes measurements of the thickness.

(14) Compute the standard deviation for the measurements of the thickness.

QUESTION #2: What is causing the spread in the measurements?

 

 

Measuring the triangles with the Calipers

(15) Now use the vernier calipers to measure the lengths a, b, and c. Record your values in the spreadsheet.

(16) Again use your spreadsheet to find d=(a2+b2)1/2 and r=c-d for the two triangles.

(17) Specify the uncertainty in your new measurements of a, b, and c by filling in the appropriate columns in your spreadsheet.

QUESTION #3: Why might the uncertainty in the measurements of a, b, and c be bigger than the uncertainty in the measurements of the thickness? Explain why you chose the numbers you chose.

(18) Given your new uncertainties, use your spreadsheet to compute Dd and Dr.

Save your file

Was Pythagoras right - Part II

(19) Now use your caliper measurements to again see if the Pythagorean theorem is true. Since the measurements are more precise than those found using the ruler, they will give a more stringent experimental test of the Pythagorean theorem.

(20) To do this, repeat the above steps 6-10 using your data from the calipers.

Question #5: Is your values for c2 using the calipers reasonable? If it is too high or low, explain what pobably is going on.

 

Homework

e-mail in a copy of your completed spreadsheet (with answers to the questions and the following homework problems) using the ftp instructions below.

1. From your results for c2, explain whether your data does or does not confirm the Pythagorean Theorem for: (a)the ruler, (b) the calipers. By the way, you won' get any credit for this question if you just say that your data does or doesn't confirm the Pythagorean theorem. What we want you to to do is start from your values for P(c2,n) and explain how they tell you that the Pythagorean theorem is or is not confirmed for each part).

2. Suppose that your ruler is marked inaccurately and is actually too short by 5%. Will this affect your conclusion about the validity of the Pythagorean Theorem? Explain.

Anonymous ftp instructions 

(This is the pedestrian way to ftp. If you are familiar with a graphical ftp like WSFTP, you can use it following the path described below).

1. From the OWL Lab menu open the windows accessories menu and select the MS-DOS prompt

2. You should see the prompt c:\user > 

3. If you spreadsheet is on a floppy type a:

4. Type ftp umdgrb.umd.edu

5. Enter anonymous for the user name

6. Then enter your e-mail address as your password

7. Change directories to phys174 (type cd phys174)

8. If you type ls You should see two directories inbox & outbox

9. Change directories to inbox (type cd inbox).

10. Type binary (If you forget this I will not be able to read your file).

11. Type put yourfile.xls (yourfile.xls should a unique name like jordan_hw1.xls If your file is named hw1.xls and you put it in the inbox it may get overwritten. You will not be able to see the contents of the inbox so you cannot remove things. You can, however, overwrite things. so if you say forgot to type binary or sent the wrong file, just resend the file with the same name).

12. Now you are done type quit

13. Remember to send an E-mail to your professor telling him you have turned in your assignment.

 

 

Spreadsheet Data Tables for Physics 174 - Triangle Exercise

Name: Lab Section: Date:

a

(mm)

Da

(mm)

b

(mm)

Db

(mm)

c

(mm)

Dc

(mm)

d

(mm)

Dd

(mm)

r=c-d

(mm)

Dr

(mm)

r2/ Dr2

Ruler triangle #1

Ruler, triangle #2

Calipers, triangle #1

Calipers, triangle #2

Results from Analysis

ruler

calipers

c2 for theory r=0

v= degrees of freedom

2

2

c2/v = reduced c2

P(c2,v)

 

 

What is P(c2,v) and how do you find it

Suppose you have found a value for c2 and that you have n degrees of freedom. Then P(c2,n) is the probability that random errors will cause c2 to be larger than you found.

Excel Tip: The easiest way to get P(c2,n) is directly from Excel by typing in the command =CHIDIST(c2,n). For example, if your value of c2 is 3 and you had five degrees of freedom, then you would type in =CHIDIST(3,5). See the help manual in Excel for more information.

Values of P(c2,n) can also be obtained from the following table.

P(c2,v)

0.99

0.95

0.9

0.70

0.50

0.30

0.10

0.05

0.01

0.001

1

0.00016

0.0040

0.0518

0.148

0.455

1.074

2.706

3.841

6.635

10.83

2

0.0100

0.0515

0.105

0.357

0.693

1.204

2.303

2.996

4.605

6.908

3

0.0383

0.117

0.195

0.475

0.789

1.222

2.084

2.605

3.780

5.423

4

0.0742

0.178

0.265

0.549

0.839

1.220

1.945

2.372

3.319

4.617

5

0.111

0.229

0.322

0.600

0.870

1.213

1.847

2.214

3.017

4.102

6

0.145

0.273

0.367

0.638

0.891

1.205

1.774

2.099

2.802

3.743

n

7

0.177

0.310

0.405

0.667

0.907

1.198

1.717

2.010

2.639

3.475

8

0.206

0.342

0.436

0.691

0.918

1.191

1.670

1.938

2.511

3.266

9

0.232

0.369

0.463

0.710

0.927

1.184

1.632

1.880

2.407

3.097

10

0.256

0.394

0.487

0.727

0.934

1.178

1.599

1.831

2.321

2.959

12

0.298

0.436

0.525

0.751

0.945

1.168

1.546

1.752

2.185

2.740

15

0.349

0.484

0.570

0.781

0.956

1.155

1.487

1.666

2.039

2.513

20

0.413

0.543

0.622

0.813

0.967

1.139

1.421

1.571

1.878

2.266

30

0.498

0.616

0.687

0.850

0.978

1.118

1.342

1.459

1.696

1.990

50

0.594

0.695

0.754

0.886

0.987

1.038

1.26

1.350

1.523

1.733

To find P(c2,n) from the table by using the following procedure:

(i) The left hand column lists n, the degrees of freedom. Find the row which has the degrees of freedom for your experiment.

(ii) Next, proceed from left to right along this row until you reach the cell which has your value of c2/n. Notice that the values increase as you go along a row from left to right. If your value of c2/n isn't listed in the table, just choose the cell in your row which has the value which is closest to your value of c2/n.

(iii) Once you have found the cell with your value of c2/n, look at the cell at the very top of this column. This is the value of P(c2,n) for your c2 and n.

For example, suppose that you have N = 9 measurements, you used one fitting parameter, and you computed c2 = 12. The degrees of freedom is thus n = N - 1 = 9 - 1 = 8 and the reduced c2 is c2/n = 12/8 = 1.5. Since n=8, we first find the n=8 row. Looking along this row, we see the values 0.206, 0.342, 0.436, 0.691, 0.918, 1.191, 1.670, etc. Since our value of c2/n is 1.5, the closest value listed in the row is 1.670. Looking at the top of this column we find the number 0.10. Thus P(c2=12,n=8) is equal to about 0.1. This means that there is about a 10% chance that random errors would produce a value of c2 which is larger than 12.