Chapter 13

Pendulum Periods

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 (A)

·    the length of the pendulum, measured from the center of the pendulum bob to the point of support (l)

·    the mass of the pendulum bob (m)

 

To investigate the pendulum, you need to do a controlled experiment; 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 capable of about 4-ms 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

CBL 2 interface

string

TI Graphing Calculator

2 ring stands and pendulum clamp

Vernier Photogate

3 masses

DataGate program

meter stick

protractor

Graphical Analysis

 

Preliminary questions

1.      Make a pendulum by tying a 100-cm string to a mass. Hold the string in your hand and let the mass swing. Observing only with your eyes, does the period depend on the length of the string? Does the period depend on the amplitude of the swing?

2.      Try a different mass on your string. Does the period seem to depend on the mass?

Procedure

1.      Use the ring stand to hang a mass from two strings (use the smallest one first). Attach the strings to a horizontal rod about 10 cm 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. 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 100 cm.

2.      Attach the Photogate to the second ring stand. Position it so that the mass blocks the Photogate while hanging straight down.  Attach the protractor to the Photogate with masking tape so that it will allow for consistent angle readings. 

3.   Connect the Photogate to the DIG/SONIC input on the CBL 2. Use the black link cable to connect the interface to the TI Graphing Calculator. Firmly press in the cable ends.

4.   Turn on the calculator and start the DATAGATE program (or app). Press  to reset the program.

5.   Set up the calculator for pendulum timing.

a.    Select SETUP from the main screen.

b.    Select PENDULUM from the PHOTOGATE SETUP screen.

 

6.   Temporarily hold the mass out of the center of the Photogate. Observe the reading on the calculator screen; you should see --0--, which indicates that the gate is open. Block the Photogate with your hand; note that the Photogate is shown as blocked (--X--). Remove your hand and the display should change to unblocked.

7.   Temporarily hold the mass out of the center of the Photogate. Select START to prepare the Photogate.

 

8.   Now you can perform a trial measurement of the period of your pendulum. Hold the mass from about 10º from vertical and release. (For a pendulum that is 100 cm long, that corresponds to pulling the bob only about 15 cm to the side.) After the calculator indicates that five trials have been recorded, press  to end data collection. The average period is displayed on the calculator screen.

9.   To return to the main screen, press . When you are ready to measure another period, select START to prepare the Photogate. You will use this method for each period measurement below.

Part I Amplitude

10.    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 30º. Each time, measure the amplitude using the protractor so that the mass with the string is released at a known angle. Repeat this step for each different amplitude. Record the data in your Data Table.

Part II Length

11.    Use the method you learned above to investigate the effect of changing pendulum length on the period. Use the lead mass and a consistent amplitude of 10º for each trial. Vary the pendulum length in steps of 10 cm, from 100 cm to 50 cm. If you have room, continue to a longer length (up to 200 cm). Repeat Steps 7 and 8 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

12.    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 Steps 7 and 8 for each mass, using an amplitude of about 10°. Record the data in your Data Table.

Data Table

Part I Amplitude                        f(A)=Τ

Amplitude (A)

Average period (T)

(°)

(s)

 

 

 

 

 

 

 

 

 

 

 

Part II Length                              f(l)=T

Length (l)

Average period (T)

(cm)

(s)

 

 

 

 

 

 

 

 

 

 

 

 

 

Part III Mass                                 f(m)=T

Mass (m)

Average period (T)

(g)

(s)

 

 

 

 

 

 

 

Analysis

1.      T is the dependent variable—on which axis is this placed? 

2.      A, l and m are independent variables.  On which axis are the independent variables placed?

3.      Using Graphical Analysis, plot a graph of pendulum period T vs. amplitude in degrees.  Use best fit line.  According to your data, does the period depend on amplitude? Explain.  

4.      Using Graphical Analysis, plot a graph of pendulum period T vs. length. Use best fit curve, power function.  Does the period appear to depend on length?  Explain.  What should the power be?

5.      Using Graphical Analysis, plot the pendulum period vs. mass. Use best fit line.  Does the period appear to depend on mass? Do you have enough data to answer this conclusively?  Explain.

6.      To examine more carefully how the period T depends on the pendulum length , create the following additional graph of the data: T 2 vs. .  Use the best fit line feature of Graphical Analysis to generate a point-slope equation for the lines. 

7.      Using Newton’s laws, we could show that for a simple pendulum the period T is related to the length and free-fall acceleration g by

, or  

         Does this last graph support this relationship? Explain.  (Hint: the term in parentheses be treated as a constant of proportionality?)  Notice the slope of the lines from the best fit line. 

What does the slope of this line represent?  What is the value of (2π)2/g?              How does the slope of your line compare to this value? 

 

 

 

Make a Word document, pasting all four graphs, data tables, and explanation/responses to questions.  Make a header with all lab members name and print copies for each member to turn in. 

 

 

 

Attach all graphs in order behind this lab procedure.