Approximate Motion of Pendulum

Theorem
Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$.

Let the bob be pulled to one side so that the rod is at a small angle $\alpha$ (less than about $10^\circ$ or $15^\circ$) from the vertical and then released.

Let $T$ be the time period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again.

Then:
 * $T = 2 \pi \sqrt {\dfrac a g}$

where $g$ is the Acceleration Due to Gravity.

Proof
At a time $t$, let:
 * the rod be at an angle $\theta$ to the the vertical
 * the bob be travelling at a speed $v$
 * the displacement of the bob from where it is when the rod is vertical, along its line of travel, be $s$.


 * MotionOfPendulum.png

From Motion of Pendulum, the equation of motion of the bob is given by:


 * $\dfrac {a^2} 2 \left({\dfrac{\mathrm d \theta}{\mathrm d t} }\right)^2 = g a \left({a \cos \theta - a \cos \alpha}\right)$

Differentiating $t$ and simplifying yields:


 * $a \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} = -g \sin \theta$

For small angles, $\sin \theta \approx \theta$ and so:


 * $\dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + \dfrac g a \theta = 0$

This is an instance of a $2$nd order linear ODE.

From Solution to Second Order Linear ODE, this has the solution:
 * $\theta = c_1 \sin \sqrt {\dfrac g a} t + c_1 \cos \sqrt {\dfrac g a} t$

We have at $t = 0$ that $\theta = \alpha$ and $\dfrac {\mathrm d \theta} {\mathrm d t} = 0$.

This yields:
 * $\theta = \alpha \cos \sqrt {\dfrac g a} t$

and so the period of this pendulum is:
 * $T = 2 \pi \sqrt {\dfrac a g}$