Simple Pendulum
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Purpose: To verify the theoretical relationship between the length of a simple pendulum and its period of oscillation. To use the simple pendulum to determine the acceleration due to gravity.

Discussion: A mass on the end of a spring is an example of a simple pendulum. If the mass is displaced from the vertical equilibrium position and then released, it will approximately execute simple harmonic motion with a period of

 where L is the length of the pendulum and g is the acceleration due to gravity. According to this formula, the period does not depend on the amplitude, but a more detailed analysis of the simple pendulum concludes that the period actually increases very slightly as the amplitude increases. Fortunately, if the amplitude is less than 0.1 radians (5.7°), the actual period should be within 0.1% of that predicted by the formula. For L = 50 cm, an amplitude of 0.1 radians corresponds to a center-to-side distance of approximately 5 cm, for L=100 cm the distance is 10 cm, and so on. You should try and keep these numbers in mind as you perform the experiment. You don't have to measure the amplitude accurately. The point is that if you keep the amplitude fairly small, then you don't have to worry about the exact value of the amplitude.

Note that if we rearrange the above formula, we can predict a linear relationship between the length L and the square of the period t,

t2= (4p2/g )L.

Procedure:

1. Attach the 200 g mass to the string, and adjust the length of the string so that L is approximately 50 cm. Carefully measure L from the top of the string to the center of the mass.

2. Carefully measure the period of small amplitude oscillations by timing 20 oscillations.

3. Repeat steps 1 and 2 for L = 60 cm, 70 cm, ..., 120 cm.  At each step you should measure the actual value of L as accurately as possible.

4. Make a graph of t2 (vertical axis) vs. L (horizontal axis). Determine the slope of your best line through the data. This slope should correspond to 4p2/g. From your slope, solve for g. Calculate the percentage error relative to the accepted value of 9.80 m/s2.

5. Make L as large as possible, and measure it carefully.

6. Determine the period of small amplitude oscillations very accurately by timing 100 (or more) oscillations. We are shooting for 0.1% accuracy in this part, so be extra careful!

7. Using the measured values of L and t in the period formula, determine g. Calculate the percentage error relative to 9.80 m/s2.

8. Can you convince yourself that the period does not depend on the mass?  Note that if you try different masses, you will have to adjust the string in order to keep L constant.