Like a Swinging Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum’s mass causes it to oscillate about the equilibrium position, swinging back and forth.

The time for one complete cycle, a left swing and a right swing, is called the period. A pendulum swings with a specific period which depends on its length. The equation of motion is:

\frac{d^2\phi}{dt^2}+\sin\phi=0

In order to solve this equation, some assuptions are needed. If the initial velocity v_0 is small, the angular displacement is small as well. Expanding \sin\phi~\phi we find:

\frac{d^2\phi}{dt^2}+ \phi=0

The general solution for the harmonic oscillator is \phi(t)=A\sin(\omega t+B). Applying the initial conditions \phi(0)=0 and \dot{\phi}(0)=v_0 we get:

\phi(t)=\phi_{max}\sin(\sqrt{ \frac{g}{l}}t)

It is important to note that the period of each oscillation T= 1/\omega=2\pi \sqrt{ \frac{l}{g}} is independent of the mass of the swinging body. This property, called isochronism, is the reason pendulums are so useful for timekeeping.

If the initial velocity is not small, the error in the Taylor expansion of \sin\phi grows larger and larger. Using advanced mathematical methods, or the conservation of mechanical energy, it can be proven that the period is given by:

T=4F(\sin\phi_{max}/2))\sqrt{ \frac{l}{g}} \sim 2\pi \sqrt{ \frac{l}{g}}\left ( 1 + \frac{1}{16}\phi_{max} + \frac{11}{3072}\phi_{max} + \hdots \right )

where F(\sin\phi_{max}/2)) is a complete elliptic integral of the first kind, that can be series-expanded as shown above.

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