Answer to Question #104050 in Differential Equations for mm

Question

If a simple pendulum of length l oscillates through an angle α on either side of the mean position then find the angular velocity dt/dθ
of the pendulum where θ is the angle which the string makes with the vertical.

Answer 1

The equation of motion for the pendulum:

$\frac{d^2\theta}{dt^2}=-\frac{g}{l}sin\theta, \;\;\theta (0)=\alpha,\;\;\frac{d\theta (0)}{dt}=0.$

For small angles $\theta$ we can use the approximation $sin\theta=\theta.$

Thus, the solution of the initial value problem

$\frac{d^2\theta}{dt}=-\frac{g}{l}\theta,\;\; \theta (0)=\alpha,\;\;\frac{d\theta}{dt}=0$

is $\theta=\alpha cos(\sqrt{\frac{g}{l}}t).$

$v(t)=\frac{d\theta}{dt}=-\alpha\sqrt{\frac{g}{l}}sin(\sqrt{\frac{g}{l}}t)$.