Lab 1:  The Simple Pendulum

Purpose

          In this laboratory you will measure the period for a pendulum of length L to see if the period is truly T= (1/2p)(L/g)1/2 .  In addition you will measure the period for different lengths to see if T µ L1/2.  This method can allow a fairly accurate determination of the local gravitational field, g.

Equipment

          You will use a ring stand, thread, a lead weight, a ruler and a timer.

Background

          A simple pendulum consists of a compact mass suspended at the end of a light string or rod of length L.  We will assume the mass of the string or rod is negligible and that we can assume all the mass of the lead weight, our compact mass, is located at the center of mass of the weight, in this case its center.  If you pull the weight to one side and release it, it will oscillate back and forth with a frequency, n, or a period, T, given by

                                                                               1.

Note that the frequency does not depend on the mass!  This expression is really an approximation that is good as long as the angle qmax is small, e.g. it is very good if qmax < 10o or 0.2 radians.  In the drawing above this corresponds to d < 0.2L.  (For d/L = 0.2, the above equation for the period is accurate to 0.25% and for d/L = 0.1 it is accurate to 0.06%.) 

          In this laboratory you will measure T and L and see if the above equation gives the correct value for g, 9.799m/s2, or 9.80m/s2 to three significant figures.  You will use the equation

                                                                                                     2.

Errors can enter into your measurements two ways. You can have errors in T and errors in L.

Experiment

          You will make a careful measurement of L and T for a pendulum of length 1.500m and see if you get the expected g, 9.80m/s2, when you use your measured L and T in equation 2.  You will repeat this for three other lengths, but in less detail.

1.     Make a simple table for your raw data, something like the spreadsheet template below.  Note that you will have two measurements for each L, one for each partner.  Don’t tell your partner what your measurement is until they have recorded theirs.

2.     Make a pendulum of length L approximately = 1.500m and measure L carefully to the nearest mm.  L is the distance from the point of suspension to the center of the lead weight.  (Again, both partners should measure L separately so you will have two measurements of L.  If they are not within 0.5cm, recheck the measurements.) 

3.     Move the lead weight aside to a position where d/L (see above) is about 0.1, i.e. make d » 0.15m if L = 1.5m. 

4.     Release the weight and measure the time for 40 complete cycles, t40.  (Later you will use this to compute the period T = t40/40.)  Do this five times to get five measurements of T with each person making at least 2 of the measurements. 

5.     Measure and record the mass of your small weight and measure the mass of the string.  Is the mass of the small weight >> than the mass of the string? 

6.     Repeat the above process for pendulums of length 0.300m, 0.600m and 1.000m.  You only need two time measurements for these lengths and you only need to measure the 1.00m time for 20 cycles.  For the two shorter lengths you might measure the time for 30 cycles and calculate the period from that.  You do not need to measure the mass again, i.e. like #5.  (Your instructor may ask you to use different lengths than the ones given above.)

         

Analysis

          Analyze this using a spreadsheet.  You want to calculate the periods, the average length and the g’s for each of the five measured periods for each length.  (Use the average L to find the g for each period.)  Then find the average g for each length, its standard deviation and the standard deviation of the mean for the average g, using equation 2.  Make a careful comparison of this calculated average g to the expected g= 9.80m/s2 for the 1.50m length.  You should note whether your calculated g is within one or two standard deviations of the expected g.  You should compare the g’s for the other 3 lengths, but you do not have a standard deviation here, so you don’t need to worry about that.  You should make a table showing the average g for each L.

In addition you should plot the average period vs. the average length.  Note that this is not a straight line.  (You do not have to include this plot in your lab.)  However, if you look at equation 1 and square it, you will see that

                                                                                                               3.

Therefore if you plot T2 vs. L it should be a straight line and the slope should be the term in parenthesis in equation 3.  Make a table of T2 vs. L and do that plot.  Also do a linear regression of T2 vs. L to find the slope.  Is the slope equal to 4p2/g?  (This means that you should see if your measured slope is within two standard deviations of 4p2/g.)

        Be sure to answer the questions below in your discussion.

1.           Is the mass of the string negligible?

2.           Are your g’s calculated from your measured L’s and T’s close to 9.80m/s2?

3.           What are the estimated uncertainties (estimated standard deviations) in your average g for the 1.50m length?  Is your measured g within two standard deviations of the expected g?

4.           Does your plot of T2 vs. L look like a straight line?  Is the slope of that line within two standard deviations of the expected slope?

Hints

I suggest that the spreadsheet be designed like the one below.  (I’ve put in particular L’s, you enter the ones you measured!)