The Simple Pendulum

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Background

The story goes that Galileo Galilei (1564 - 1642) was in Church one day when he found his attention diverted from the Mass to the regular swaying of a lantern. Using his pulse as a crude watch, he observed that the period of the motion was independent of its amplitude, an observation he later sought to confirm with greater precision using a water clock. He soon hit upon the idea of using the steady motion of a pendulum to regulate a mechanical clock, although the first workable design of such a mechanical clock was executed (and patented) by the Dutchman Christiaan Huygens (1629 - 1695). The pendulum clock became the standard time-keeping apparatus of Europe and was refined and improved for over two centuries, before being overtaken by spring-based timekeepers (watches), quartz oscillators, and atomic clocks. In this experiment, you will investigate the behavior of pendulums to discover what properties of a pendulum influence the period of its swing. Unlike Galileo and Huygens, you will have the benefit of rather precise timing apparatus (Photogates and Timers), with which you can make detailed and accurate observations of the pendulum's motion. In particular, you can (read: "you better!") avoid Galileo's mistake of thinking that the period does not depend on the amplitude of motion.

Theory

A simple pendulum consists of a point-like bob suspended from a massless, inextensible string from a fixed point, as illustrated in the figure. Released from rest at an angle q_o from the vertical, the bob descends in a circular arc, rises again to the same amplitude on the opposite side, and then reverses course to return to the starting position, providing that friction is negligible. If not, the motion gradually decays until the bob comes to rest vertically below the point of suspension.

To analyze the motion one can consider forces on the bob, torques on the bob about the point of suspension, or energy conservation of the bob during the motion. This last method lends itself well to numerical approximation on a computer. More on that below.

The pendulum bobs you will use (at least initially) are metal cylinders. (The light ones are made of aluminum; the heavy ones of iron.) They are not really point masses, although to a reasonable approximation they behave as point masses located at the center of mass of the cylinder. Therefore, the length L of the pendulum is as shown in the figure.

Apparatus

• photogate and mini-module timer circuits
• aluminum, iron, and plastic bobs
• nylon fishing line
• protractor
• meter sticks and rulers

Possible Experiments

Your first job is to figure out how the timing apparatus works:

• What does the circuit time and how? You can use your finger passing through the photogate to learn how it is tripped, when it starts, when it stops, etc. Check the times it records against your wristwatch to make sure you understand the readout. Look for the clock circuit (one of the mini-modules) to see what timing options you have.
• How repeatable are the measurements and how can you minimize trial-to-trial variations? What are the principal sources of error?

When you are confident you understand these, use Dimensional Analysis to learn theoretically as much as you can about the dependence of the period on the parameters of the motion (length, amplitude, mass, ... ). Be sure to put this calculation in your laboratory notebook. Then discuss with your partner in which order you would like to investigate the various dependencesof the period. Decide on an ordering, then call one of the professors over to discuss it with him.

Remember to keep detailed notes of the procedure you follow, records of your data, plots of your results, analysis, etc., in your laboratory notebook. At the end of the four days of experimentation you will be writing a formal analysis of the behavior of the pendulum, in which you will be using some of your data and the plots you have made. See the Reports page for further information on the report. Please be sure that each partner keeps a copy stored in the proper placed on the course account: KATO.HOME/Physics/Ph23A/yname/pendulum/

Tips and Warnings

Take care in releasing the pendulum bob to avoid hitting the timing gate.

For very small q_o, the bob may not clear the photogate to trigger the timer. In such cases, you can move the photogate a bit to one side so that one side of the bob moves in and out of the beam. How will this affect the timer?

Modeling

For some ideas on how to model the amplitude dependence of the period of the pendulum, see this paper.

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Updated 9/3/00 by Peter N. Saeta .