Experimenting With Compound Pendulums And Moment of Inertia.

December 1, 2003

Experimenting With

Compound Pendulums

And Moment of Inertia

Abstract

Two compound pendulums with different weight distributions were used to experimentally determine if the laws of simple harmonic motion would apply to them as well. The moments of inertia were determined experimentally, based on the periods of the pendulums, and compared to theoretical calculations. The average percent error for the pendulum with the shorter R (distance from pivot point to cm of the weights) was 2.67%, and for the longer R was 6.15%.

Introduction

The laws of simple harmonic motion are based on the periodic displacement, acceleration, and velocity of an object. A period is the time taken between peaks of maximum amplitude. When this type of motion is free of nonconservative forces, and the force needed to displace the object is proportional to the displacement, it is called simple harmonic motion.

A simple pendulum is one in which a point mass is suspended from a string of negligible mass. It swings with a period of:

T = 2π√(L/g)

where L is equal to the length of the pendulum. In a compound pendulum, the mass of the pendulum arm provides torque and must be described using Newton’s second law for rotation:

τ = Iarmα, and the torque is τ = -mgLcmsin θ. This leads to the equation for the period of a compound pendulum:

T = 2π√(Iarm/mgLcm)

which is similar in form to the simple pendulum equation, adding the compensation for the moment of inertia inherent in the arm of the pendulum.

A pendulum is suitable for experiments in simple harmonic motion, because it provides a motion similar to a vertical spring oscillating up and down (or back and forth for a horizontal spring). This is because gravity provides the force to move the pendulum initially from its point of all potential energy to its point of all kinetic energy. In the absence of friction and air resistance, a pendulum would oscillate forever in simple harmonic motion, much the way an ideal spring would.

In this experiment, a compound pendulum was constructed out of two weights that could be screwed together through a series of holes in a flat steel bar. The weights were attached near one end, and the center of mass of the system was determined by balancing. A hole toward the other end of the bar was chosen as the pivot point, and measurements were taken for the length of the bar, the width of the bar, the center of mass of the bar to the pivot point, the center of mass of the pendulum to the pivot point, the center ...

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The pivot hole was placed over a knife-edge support, and the pendulum was pulled to the side and released to start it oscillating. One oscillation is the motion for one complete trip to and fro. The swing needed to be less than ten degrees from vertical, so that sinθ ≈ θ. A stopwatch was used to measure the time for ten complete oscillations. Five trials were performed, and the experimental moment of inertia was calculated from the resulting period.

The weight was moved on the bar, and a different pivot point was chosen for a second set of trials. The resulting moments of inertia were then compared to theoretical calculations for the moment of inertia.

Purpose

To demonstrate that the laws of simple harmonic motion apply to a compound pendulum.

Procedures

Take measurements of the mass of the bar, the mass of the weights, the length and width of the bar, the radius of the weight, and distances from the pivot point to the centers of mass of the bar, the weight, and the bar and weight combined.
Select a pivot hole. Hang the bar from the knife edge support through the pivot hole and swing the pendulum with an angle of less than ten degrees.
Start timer at the beginning of an oscillation and stop it after ten complete oscillations.
Record the time.
Repeat for five trials.
Change conditions, i.e. the location of the weight on the bar and the pivot hole used, and repeat the above.

Sources

Wozniewski, L. (2000). Physics Laboratory Manual: Coefficient of Static and Kinetic Friction. Retrieved October 19, 2003, from Indiana University Northwest, Department of Chemistry, Physics, and Astronomy Web site: http://www.iun.edu/~cpalw/pweb/pendulum/pendulum.htm

Cutnell, John and Johnson, Kenneth. Physics Sixth Edition. Hoboken, NJ: Wiley and Sons, 2004.

Tables of Experimental Data

Sample Calculations

Results

Conclusion

The motion of a compound pendulum was similar to the motion of simple harmonic motion. The oscillations of the pendulum were similar in velocity, acceleration, and period to that of an ideal spring oscillating back and forth in harmonic motion. Therefore, the laws of simple harmonic motion could be applied to a compound pendulum also, because the experimental moment of inertia was comparable to the theoretical moment of inertia.

It should be noted that moving the center of mass of the weights further from the pivot point increased the moment of inertia, and thus slowed down the rotation of the pendulum about the pivot point. Bringing the weight in closer to the pivot point provided less torque, which allowed faster times for the period.

The average percent error for the shorter R was 2.67%, and for the longer R was 6.15%. This may be due to the greater time involved in the swing, which allowed friction and air resistance more time to act on the pendulum, as well as the larger swing area that front to back movement of the pendulum could have been introduced. Also, the theoretical calculation was based on ideal conditions, without friction or air resistance, which could not be obtained in the lab setting.

Human error could have played a role in the error of the experiment, also. Measurements were taken by yardsticks, and the angle that the pendulum was started at may have been greater than ten percent, which could have skewed the results on one or both of the pendulums.