The Simple Pendulum

The simple pendulum consists of a "point" mass attached to a massless string. In reality, all pendulums are physical pendulums and a knowledge of the pendulum's moment of inertia about the axis of oscillation must be known. However, our strings have negligible mass and the size of the weights are small compared to the length of string. In this lab we will experimentally determine how the period is related to the mass, the length, and the amplitude.

A. Periodicity and Mass

Using the triple beam balance, find the mass of the pendulum screweye assembly. Install one of the 100 gram masses into the screweye assembly. The total mass of the simple pendulum includes the assembly. Attach the assembly to one end of a string of length approximately 2 meters. The distance should be measured accurately from the support point to the center of the 100 gm mass. (The assembly is set up such that the c.o.m. of the system is very near the geometrical center of the masses.) Pull the pendulum out 5 to 10 degrees and release. Time the pendulum for 10 cycles and find the period.

Repeat three more times, adding an additional 100 gram mass each time. In each case, you must adjust the string so that the distance from the support point to the center of the masses remains the same. Be sure to always release from the same angle.

• Plot period vs mass. For clarity, set the period axis minimum to zero for this plot.
• How does the period depend upon the mass of the pendulum? Explain why this is so?

B. Periodicity and Length

With two or three 100 gm masses installed, find the period of the pendulum with lengths of approximately 2.0, 1.75, 1.50, 1.25, 1.00, .75, and .5 meters. It is not necessary to use exactly these values, but it is necessary to measure them accurately. Keep the amplitude angle small and the same for each trial.

• Plot the period vs the length.
• Plot the square  of the period vs the length and fit it with a straight line.
• Consider the theoretical formula for the period of a pendulum.
  T = 2p (^L/_g)^1/2

  From this formula, explain why the data in the second plot should fall along a straight line.

• From the formula, what should be the theoretical slope of the line in the second plot? Compare the actual slope of the line in the second plot with this theoretical value.

C. Periodicity and Amplitude

Simple harmonic motion has the property that the period is independent of the amplitude. Adjust the pendulum to a length of about 1 m. Determine the period as before, for amplitudes of 10, 20, 30, 40, 50, 60 and 70 degrees.

• Plot the period vs amplitude and interpolate a smooth curve through the points. For clarity, make two plots; one with zero for the minimum value of the vertical axis and one with the value of the period at 10 degrees as the minimum value.
• For a simple harmonic oscillator, the period should be the same for all amplitudes. What does your data tell you about the simple pendulum?
• Consider how the period changes as amplitude increases. Would you conclude that the restoring force acting on the pendulum increases at a greater rate than a linear restoring force or at a lesser rate? Give qualitative arguments why this so.