Question

A pendulum with a time period of 1 s is losing energy due to damping. At a certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, find its damping constant.

Answer 1

We know that, the amplitude of the damped oscillations at any time $t$ is given by,

$A=A_0{e}^{-\frac{bt}{2m}}$
So, energy at any time $t$ is given by, E$=\frac{1}{2}K{A}_0^2{e}^{-\frac{bt}{m}}$
Given, Time = $t=15s$

Energy at $t=0$ s = 45 J.

Energy at $t=15$ s = 15 J.
Therefore,

$15=45e^{-\frac{b15}{m}}$
$\Rightarrow \frac{1}{3}=e^{-\frac{b15}{m}}$
$\therefore \frac{b}{m}=\frac{1}{15}ln3$.