A particle is executing simple harmonic motion (SHM). The ratio of...

The formula to calculate the potential energy of the particle executing SHM is given by

$P = \dfrac{1}{2}m{\omega}^{2}{x}^{2} . . . \left(1\right)$

The formula to calculate the kinetic energy of the same particle is given by

$K = \dfrac{1}{2}m{\omega}^{2}\left({A}^{2} - {x}^{2}\right) . . . \left(2\right)$

Divide equation (1) by equation (2) to obtain the required ratio.

$\begin{cases}\dfrac{P}{K} & = & \dfrac{\dfrac{1}{2}m{\omega}^{2}{x}^{2}}{\dfrac{1}{2}m{\omega}^{2}\left({A}^{2} - {x}^{2}\right)} \\ & = & \dfrac{{x}^{2}}{{A}^{2} - {x}^{2}} . . . \left(3\right)\end{cases}$

Substitute $\dfrac{A}{2}$ for $x$ into equation (3) to calculate the required ratio.

$\begin{cases}\dfrac{P}{K} & = & \dfrac{{\left(\dfrac{A}{2}\right)}^{2}}{{A}^{2} - {\left(\dfrac{A}{2}\right)}^{2}} \\ & = & \dfrac{1}{3}\end{cases}$