Question

Find an expression for energy of a body vibrating in SHM.

Answer 1

Let the SHM equation be

$X=Asin\omega t$

Now, $\frac{dx}{dt}=A\omega cos\omega t=v$

$\frac{d^{2}t}{d{t}^2}=-A{\omega }^{2}sin\omega t=a$

Potential energy is given by

$=1/2k{x}^2=1/2m{\omega }^2{k}^2$

$=1/2m{\omega }^2{A}^{2}si{n}^2\omega t$

Kinetic energy = 1/2 m$v^2$

$=1/2m{\omega }^2{A}^{2}con{s}^2\omega t$

Total energy = PE + KE

$1/2m{\omega }^2{A}^{2}si{n}^2\omega t+1/2m{\omega }^2{A}^{2}co{s}^2\omega t$

$=1/2m{\omega }^2{A}^2$