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:
This is almost a simple harmonic equation with
. To make it one, we have to use the small angle
If you compute the gravitational potential energy for the pendulum for
arbitrary angle , you get:
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.