Question

Obtain an expression for the kinetic energy of oscillation and potential energy of a body performing a simple harmonic motion. Show that the total energy of a particle executing simple harmonic motion is proportional to the square of the amplitude and frequency of vibration.

Answer 1

Let the equation of SHM be
$x=Asin\omega t$
$\Rightarrow v=A\omega cos\omega t$
Kinetic energy, $K=\frac{1}{2}m{v}^2=\frac{1}{2}m{A}^2{\omega }^{2}co{s}^2\left(\omega t\right)$
Potential energy, $V=\frac{1}{2}k{x}^2=\frac{1}{2}K.A^{2}si{n}^2\omega t=\frac{1}{2}m{\omega }^2{A}^{2}si{n}^2\omega t$
Total energy, $E=K+V=\frac{1}{2}m{w}^2{A}^2(co{s}^2\omega t+si{n}^2\omega t)$ $[\therefore k=m{\omega }^2]$
$E=\frac{1}{2}m{\omega }^2{A}^2$
$\therefore E\propto {\omega }^2{A}^2$