Question

When the displacement in S.H.m. is $\frac{1}{3}rd$ of the amplitude . what fraction of total energy is kinetic energy and what fraction is potential energy?

Answer 1

$T.E=\frac{1}{2}m{w}^2{a}^2$
$K.E=\frac{1}{2}m{w}^2(a^2-n^2)$
$n=\frac{1}{3}a,K.E=\frac{1}{2}m{w}^2(a^2-\frac{1}{9}{a}^2)$
$K.E=\frac{1}{2}m{w}^2{a}^2\left(\frac{8}{9}\right)$
Now $\frac{K.E}{T.E}=\frac{8}{9}$
$P.E=\frac{1}{2}m{w}^2{w}^2{n}^2$
$P.E=\frac{1}{2}m{w}^2\left(\frac{1}{9}\right)a^2$
$\frac{P.E}{T.E}=\frac{1}{9}$