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## The Pendulum

The pendulum is another example of a simple harmonic oscillator, at least for small oscillations. Suppose we have a mass attached to a string of length . We swing it up so that the stretched string makes a (small) angle with the vertical and release it. What happens?

We write Newton's Second Law for the force component tangent to the arc of the circle of the swing as:

 (70)

where the latter follows from (the angular acceleration). Then we rearrange to get:
 (71)

This is almost a simple harmonic equation with . To make it one, we have to use the small angle approximation . Then

 (72)

and we can just read off the solution:
 (73)

If you compute the gravitational potential energy for the pendulum for arbitrary angle , you get:

 (74)

Somehow, this doesn't look like the form we might expect from blindly substituting into our solution for the SHO above:
 (75)

As an interesting and fun exercise (that really isn't too difficult) see if you can prove that these two forms are really the same, if you make the small angle approximation for in the first form! This shows you pretty much where the approximation will break down as is gradually increased. For large enough , the period of a pendulum clock does depend on the amplitude of the swing. This might explain grandfather clocks - clocks with very long penduli that can swing very slowly through very small angles - and why they were so accurate for their day.

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Next: The Physical Pendulum Up: Oscillations Previous: Energy   Contents
Robert G. Brown 2004-04-12