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) |

(71) |

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

(72) |

(73) |

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

(74) |

(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.