LAB 1:  MEASUREMENTS AND UNCERTAINTIES

 

PURPOSE:

      In this lab I want you to think about how experiments test theories, to learn about the uncertainties involved in measurements and to learn how to make and interpret simple graphs.  I also want you to use a spreadsheet to analyze the data from this laboratory. 

      At the end of the lab you should know the meaning of standard deviation, mean value, standard deviation of the mean, linear regression and the concept of uncertainty. 

 

EQUIPMENT

      Four disks of different diameters, meter stick, graph paper and ruler.

 

INTRODUCTION

      Physics is an empirical science, and theories are judged by how well they agree with experiment.  In an elementary Physics course it is easy to get the opposite impression because we spend most of the time talking about theo­ries.  (One reason for this is that good experiments are often time con­sum­ing, require good equipment and careful design.  As a result it is often easier to talk about theories than to do or even explain the ex­periments that led to the theories.)

      When we make measurements to test theories, it is necessary to know how reliable or accurate the measurements are.  For instance, in a flat space one would expect the area and the circumference of a circle to be related to the radius by

          Area =  pr².                                                                                                                    1.

and

          Circumference = 2pr                                                                                                      2.

where  p = 3.1416 approximately.  However, what would happen if our 'circles' were not really flat, but sections on the surface of a sphere with a very large radius, R?  Then our areas and circumferences would not be given by equations 1 & 2.  They would work very well for r << R, but they would only be approximations and would not be accurate at large r's. 

      We could test the theory that the earth is flat by measuring the radii and circumferences of large circles on the earth's surface and seeing whether they agree with equation 2.  One could also make predictions based on a spherical earth and compare our measurements with them.  The diffi­culty is that we would have to use large r's (not always practical), or make very accurate measurements to distinguish between the two theories.  For the surface of the earth and an r = 10 km we would have to measure our distances to about one part in ten million (1 mm) to distinguish between the predictions of the two theories.  (Can you develop a theoretical relationship between the radius and circumference for "circles" that are on the surface of a large sphere of radius R?)

      The main points of this discussion are that experiments are the final judge of a theory and that an experiment can only test a theory to the accu­racy of the experiment itself.

      Just how accurate is a given measurement?  We will look at this question as we try to test eqn. 2 for four disks of different diameters.  For each disk, we will measure the diameter and the circumference.  We will estimate the accuracy of these measurements and compare the relationship between the diameter and the circumference to equation 2.  The experiment itself is trivial; we want you to learn about some basic sta­tistical ideas and about how to present and analyze data. 

      There is another consideration in making a measurement.  Are you measuring what you intended to measure?  Your samples and measurement conditions are not always what you think they are and this can limit the validity your measurements!  For example if you wanted to measure the density of silver, you would want a sample of pure silver.  If you used a silver dollar  made of "coin" silver, which is 90% silver and 10% copper, it would not have the same density as pure silver.  For this lab it may be that your disks are not true circles. 

 

PROCEDURE

1)       If you are not familiar with the metric system, please read your text­book.  We will use the metric sys­tem in our labs. 

2)       Look at the meter stick on your table; the finest divisions marked on it are millimeter, mm, divisions.  With care, we can measure about half the distance between divisions, or 0.5mm.  Therefore, assume the accuracy of a measurement is no better than ± 0.25mm and only measure to the nearest 0.5mm.

4)      Measure the diameter and the circumference of a disk, one partner measuring, the other recording.  The diameter should be measured through the center of the disk.  To measure the circumfer­ence, make a mark on the edge of the disk and roll the disk along the meter stick for one revolution to obtain the circumference.  Try to keep the disk from slipping. 

5)      Recorder and measurer should exchange jobs and repeat the mea­surement for the same disk.   A total of ten readings should be made of the diameter and circumference, with partners trading jobs between each set of measurements.  Do not measure the diameter in the same location for all trials, but mea­sure it across various portions of the disk, making sure each mea­surement goes through the center.  Record them in a table like the one below.

	Raw Data
Trial	Circum (cm)	Dia (cm)
1
2
3
4
5
6
7
8
9
10

 

6)      Now that you have ten measurements of the diameter and circumference for each disk, sit back and savor your data.  Do you notice minor fluctuations in the values?  Which ones are correct?  They all are legitimate measure­ments.  Our best estimate of the true value is the av­erage or mean.  Determine the average value of the diame­ter and the circumference.  To calculate these enter your raw data into a spreadsheet.  The data for each disk should look like the one shown in the supplement.  Have the spreadsheet calculate the average circumference and average diameters below the appropriate column using the AVERAGE(..) function.

7)      How reliable are these average values? The standard deviation, s, is an estimate of the uncertainty due to random fluctuations in your measurements.  It is a measure of how the individual measurements fluctuate about the average, or mean, value.  If the fluctuations in the measurement of the quantity are due to random errors, one would expect an individual measurement of that quantity to have a 68% probability of being within ± one standard deviation of the mean value.  It has a 95% probability of being within ± two standard deviations of the mean value.  (This is only true for Gaussian or Normal distributions, but we won’t worry about those details.) 

 

NOTE:  To calculate the standard deviation "by hand", you would first calcu­late the deviation (difference) of each of your measurements from the average and the squares of those deviations.  See the sample data table below.

Trial                  Diameter                        Deviation                        (Deviation)²

  1                         10.25 cm                      +    .01 cm                        .0001 cm²

  2                         10.20                            -     .04                             .0016

  3                         10.30                            +    .06                             .0036

  4                         10.25                            +    .01                             .0001

  5                         10.30                            +    .06                             .0036

  6                         10.20                            -     .04                             .0016

  7                         10.10                            -     .14                             .0196

  8                         10.15                            -     .09                             .0081

  9                         10.35                            +    .11                             .0121

         AVE =                                              10.24 cm         SUM =    .0540 cm²

       The average diameter = 10.24 cm and the sum of (deviations)² = 0.054 cm² .  The square of the standard deviation squared, s², is given by the relation

       In the above table, the third column lists the square of each deviation and the sum = .0540. Dividing this by 9 and taking the square root gives a standard deviation of 0.077cm.  This is an estimate of the uncer­tainty of an INDIVIDUAL measurement due to random fluctuations. 

      The Spreadsheet can calculate these for you, you do not need to go through the steps above.  You can calculate the standard deviations for the circumference and diameter using the STDEV(..) function in Excel. 

               The uncertainty for the MEAN of n measurements is less than the uncertainty for an individual measurement.  The best estimate of the standard deviation for the mean of n measurements is

         or for the example above, s_m = 0.077/3.16 = 0.024 cm.  Then our best estimate of the diameter is the mean value and the uncertainty is + s_m, (some people use + 2s_m), or

            diameter = 10.24 cm ± .024 cm                                                                                 5.

         This means the diameter is expected, with a probability of 0.68, to be between 10.00 and 10.48 cm.  The probability is 0.95 that it is within 2 standard deviations of the mean or between 9.76 and 10.72 cm. Use a spreadsheet to calculate the standard deviations for the circumferences and diameters of all four disks.  Also find the standard deviations of the mean diameter and circumference of each disk.

9)      For each disk you should estimate p  by taking the mean cir­cumference divided by the mean diameter.  Also find the percent difference between each p and the accepted value of p = 3.1416.  (See #12 below for the percent difference.)  The uncertainty in your estimate of p can be calculated from the uncertainties, standard deviations, for the mean circumference and mean diameter.  Estimate the uncertainty in  p, s_p.  Are your cal­culated values of   within ± s_p  of the accepted/predicted value of  p?  If your mea­sured value is within ± 2 s   of p , then there is little or no statistically significant differ­ence between your measured value and the predicted value.  Is there a significant difference between your p’s and the accepted value of p?

                 Note that if you have c = a/b and you know the standard deviation of a and b, s_a and s_b, you can estimate the standard deviation for c, s_c in the following way. 
It is the fractional uncertainties, s_a/a, s_b/b and s_c/c that are used to compute this.  It turns out if c = ab, a times b, the standard deviation for c is calculated the same way!  This works as long as the fractional uncertainties are much less than 1.

10)    Now examine the relationship be­tween the diameter and the circum­ference by plotting the aver­age value of circumference versus the average value of diameter on graph paper.  (I recommend you do this in a spreadsheet.)  The Diameter should be plotted on the x-axis and the circumference on the y-axis.  Spread out the data so that you get maximum resolu­tion.  You should make a simple table of the mean diameters and mean circumferences.  Ask for help if you are confused.  We should be able to draw a straight line through our data points.  This straight line means that there is a direct, linear relationship between the variables we have graphed.  In other words, the circumference and diameter are directly proportional and the constant of proportionality should be p.  Algebraically this means:

            (circumference) = (constant) * (diameter)                                                                   6.

         If you remember high school geometry, this constant is called the slope.  (We will use the symbol m for the slope.)  The equation of a straight line is

            y = mx + b                                                                                                                   7.

         In our case  y = circumference, m should be p and x is the diameter.  (What should b equal?)

11)    Determine the value of the slope (p) by using the linear regression routine in the spreadsheet.  Don't forget the units.  This value should be about the same as the average of the four p's you calculated in #9 (for the four different disks).  Is it?

12)    Determine the percent difference between your slope and the gen­erally accepted value 3.1416.  The percentage difference is

         Note that a negative percentage difference means your result is less than the accepted value.  

 

Questions:

1.      How accurate are your results and what factors limit the accuracy of your results?  Try to be quantitative.  (Are your results more than 2 standard deviations from the generally accepted value of p?)

2.      Does the straight line fit (linear regression) of mean circumference vs. mean diameter go ‘through’ the origin, i.e. have an intercept of zero?  Should it?  Why ?

3.      Do you feel that your disks were "true" circles?  Why?

 

Layout

I recommend the layout at the right for your data and computations.  A good layout makes it much easier to examine the data and compare the results to other people’s data.  The sample shows the data and computations for one disk.

Circum Trial dia (cm)  (cm) 1     2     3     4     5     6     7     8     9     10     mean     Std Dev     Std Dev mean     Pi     Std Dev Pi