A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many years the pendulum was the heart of clocks used in astronomical measurements at the Greenwich Observatory.

There are at least three things you could change about a pendulum that might affect the period (the time for one complete cycle):

the amplitude of the pendulum swing

the length of the pendulum, measured from the center of the pendulum bob to the point of support

the mass of the pendulum bob

To investigate the pendulum, you need to perfrom a controlledexperiment; that is, you need to make measurements, changing only one variable at a time. Conducting controlled experiments is a basic principle of scientific investigation.

In this experiment, you will use a Photogate/Motion Detector capable of microsecond precision to measure the period of one complete swing of a pendulum. By conducting a series of controlled experiments with the pendulum, you can determine how each of these quantities affects the period.

Figure 1

objectives

Measure the period of a pendulum as a function of amplitude.

Measure the period of a pendulum as a function of length.

Measure the period of a pendulum as a function of bob mass.

Materials

Power Macintosh or Windows PC	String
Universal Lab Interface	2 ring stands and pendulum clamp
Logger Pro	masses of 100g, 200g, 300g
Vernier Photogate or Motion Detector	meter stick
Protractor	Graphical Analysis or graph paper

Procedure

1. Use the ring stand to hang the 200-g mass from two strings. Attach the strings to a horizontal rod about 10cm apart, as shown in Figure 1. This arrangement will let the mass swing only along a line, and will prevent the mass from striking the Photogate/Motion Detector. The length of the pendulum is the distance from the point on the rod halfway between the strings to the center of the mass. The pendulum length should be at least 1 m.

2. Attach the Photogate to the second ring stand. Position it so that the mass blocks the Photogate while hanging straight down. Connect the Photogate to the DG 1port on the Universal Lab Interface.

3. Prepare the computer for data collection by opening "Exp 14" from the Physics with Computersexperiment files of Logger Pro. A graph of period vs. measurement number is displayed.

4. Temporarily move the mass out of the center of the Photogate. Notice the reading in the status bar of LoggerProat the bottom of the screen, which shows when the Photogate is blocked. Block the Photogate with your hand; note that the Photogate is shown as "blocked." Remove your hand, and the display should change to "unblocked." Click and move your hand through the Photogate repeatedly. After the first blocking, Logger Proreports the time interval between every other block as the period. Verify that this is so.

5. Now you can perform a trial measurement of the period of your pendulum. Pull the mass to the side about 10 from vertical and release. Click and measure the period for five complete swings. Click . Click the Statistics button, , to calculate the average period. You will use this technique to measure the period under a variety of conditions.

Part I Amplitude

6. Determine how the period depends on amplitude. Measure the period for five different amplitudes. Use a range of amplitudes, from just barely enough to unblock the Photogate, to about 30. Each time, measure the amplitude using the protractor so that the mass with the string is released at a known angle. Repeat Step 5 for each different amplitude. Record the data in your data table.

Part II Length

7. Use the method you learned above to investigate the effect of changing pendulum length on the period. Use the 200-g mass and a consistent amplitude of 20 for each trial. Vary the pendulum length in steps of 10cm, from 1.0 m to 0.50 m. If you have room, continue to a longer length (up to 2 m). Repeat Step 5 for each length. Record the data in the second data table below. Measure the pendulum length from the rod to the middle of the mass.

Part III Mass

8. Use the three masses to determine if the period is affected by changing the mass. Measure the period of the pendulum constructed with each mass, taking care to keep the distance from the ring stand rod to the center of the mass the same each time, as well as keeping the amplitude the same. Repeat Step 5 for each mass, using an amplitude of about 20. Record the data in your data table

Data Table

Part I Amplitude

Amplitude ()	Average period (s)
0	0
5	2.030
10	2.040
15	2.050
20	2.049
25	2.043

Part II Length

Length (m)	Average period (s)
0	0
0.5	1.431
0.6	1.565
0.7	1.690
0.8	1.808
0.9	1.919
1	2.039

Part III Mass

Mass (g)	Average period (s)
0	0
100	2.039
200	2.049
300	2.064

Analysis

1. Why is Logger Proset up to report the time between every otherblocking of the Photogate? Why not the time between every block?

2. Using either Graphical Analysis or graph paper, plot a graph of pendulum period vs. amplitude in degrees. Scale each axis from the origin (0,0). Does the period depend on amplitude? Explain.

3. Using either Graphical Analysis or graph paper, plot a graph of pendulum period T vs. lengthl. Scale each axis from the origin (0,0). Does the period appear to depend on length?

4. Using either Graphical Analysis or graph paper, plot the pendulum period vs. mass. Scale each axis from the origin (0,0). Does the period appear to depend on mass? Do you have enough data to answer conclusively?

5. To examine more carefully how the period T depends on the pendulum length l, create the following two additional graphs of the same data: T 2vs. land T vs. l 2. Of the three period-length graphs, which is closest to a direct proportion; that is, which plot is most nearly a straight line that goes through the origin?

6. Using Newton's laws, we could show that for some pendulums, the period Tis related to the length land free-fall acceleration gby

, or

Does one of your graphs support this relationship? Explain. (Hint: Can the term in parentheses be treated as a constant of proportionality?)

7. From your graph of T² vs. l, determine a value for g.