Course and Section_______                            Names ___________________________

 

Date___________                                           _________________________________

 

Error Analysis Experiment

 

 

In this experiment we are interested in learning how to treat data.  We will study a mass oscillating on a spring in order to illustrate the concepts that are important.  The main point here is that any time a scientist measures a number,  she must be able to tell her colleagues the accuracy of that number.  Otherwise, there is no way to tell whether the number agrees with the predictions of a theory; there is also no way for another scientist to check the experiment.  For example, suppose I measure the circumference of a circle, then its diameter, and divide the circumference by the diameter.  The result ought to be p.  If my result is 3.15, have I proved that p is not 3.14???  In this case, of course, we know the accepted result.  If the uncertainty of my measurement is 0.01 or more, then my result is consistent with the value that we are familiar with.

 

Your apparatus is a spring with a mass holder hanging from it.  You are to measure the period of the spring, which is the time it takes for the mass to make a complete journey from top of its trip, to the bottom, and back to the top.  The best way to measure this period is to time 5 oscillations, and then divide by 5 to get the time for one oscillation.  Make tables to record your data as you do the following. 

 

1.  Place about 20 grams on the mass holder; set it into oscillation and time its period as described above.  In order to see how results can vary, repeat this a total of 5 times.  You won’t have to do this in the remainder of the experiment.  Record all 5 measurements, and manually calculate the average of these times.  Record this average as well.

 

2.  Now add 20 grams to the mass holder and repeat the procedure.  This time, however, you only need to make one measurement, not 5.  Do this until you have 8 sets of measurements, corresponding to 8 different masses.

 

Analysis:

 

A result that you will learn later in the semester shows that the square of the period is proportional to the mass:

                                            (1)

 

We wish to find the best value for the proportionality constant, “B.”

 

Method 1.  Graphical estimate.

 

First add a column to your data table(s), calculate T2 for each of your masses, and write the value in this new column.  On graph paper, plot the 8 data points, T2 vs. m.  Now take a ruler and draw the most reasonable line that averages out your data points; use your best judgment to find this line.  Next draw another line that is steeper, but still is barely consistent with your data; finally draw a line that is less steep yet could be said to be barely consistent with your data.  Your instructor will indicate just how to do this.

 

Find the slopes of each of your three lines, and write these slopes down as Bo, the slope of the “best” line, Bhi, the slope of the steepest line, and Blo, the slope of the least steep line. Report your final result as

           

                                                                                       (2).

 

dB is the uncertainty in your measurement of B, and is calculated as the average difference between your best slope and the "extreme" slopes.  

 

dB = ½ {(Bhi – Bo) + (Bo – Blo)} = ½ (Bhi – Blo).         (3)

 

You now have an estimate of the value for B, and also an estimate of the uncertainty (dB) in that value.  This estimate was not made by any mathematical formula, but rather by judging by eye the range of lines that might represent your data.  Next we will use some mathematical approaches.

 

 

 

Method 2:  Linear Regression.

 

Note:  This part may be assigned as homework.

 

The method of linear regression uses a mathematical procedure to find the line which best represents a set of data.  “Best” in this case means minimizing the difference between the line and the data.  The line will be represented by

 

                                                            y = A + Bx.                              (3)

 

Without trying to derive the result, we will just tell you that according to linear regression, the best value for the slope of a set of data points is given by

 

,                   (4)

 

where

.                   (5)

 

In this set of formulas, the “x” values are the data values of the variable normally plotted along the “x” axis; in this experiment, this is the mass.  The other variable, y, is the corresponding value of T2 in this example.  N is the number of data points (8, in this case).

 

You can also find the best value for A:

 

.           (6)

 

 

 

Use these formulas to find the best values for the slope B and the intercept A.  To find the uncertainty dB, we first must find the average variation between the line and the data points.  The best estimate of this is:

 

.                  (7)

 

It then turns out that the uncertainty in B is

 

.                         (8)

 

 

Use the above formulas to find the uncertainty in B.  Compare your results to the ones you found by “eyeball” using Method 1.

 

Method 3.  Calculators.  (Optional)

 

Use either your calculator or your computer (for example, use Excel) to find the line that best fits your data points, and the uncertainty in this line.  Again, compare this result with those found earlier.  Comment on your observations.