Consider a particle of mass m executing S.H.M. with period T. The displacement of the particle at an instant t, when time period is noted from the mean position is given by $y = A \sin \omega t$ ∴ Velocity, $v = \frac{dy}{dt} = A \omega \cos \omega t$ $\text{K.E.},\, E_{k} = \frac{1}{2} m v^{2}$ $= \frac{1}{2} m A^{2} \omega^{2} \cos^{2} \omega t$ P.E., $E_{p} = \frac{1}{2} k y^{2}$ $= \frac{1}{2} m A^{2} \omega^{2} \sin^{2} \omega t$ $(\because k = m \omega^{2})$ ∴ Average K.E. over one cycle $= \frac{1}{2T} m A^{2} \omega^{2} \int\limits_{0}^{T} \frac{1+\cos 2\omega t}{2} \, dt$ $= \frac{1}{4T} m A^{2} \omega^{2} \left[ t + \frac{\sin 2\omega t}{2\omega} \right]_{0}^{T}$ $= \frac{1}{4T} m A^{2} \omega^{2} (T) = \frac{1}{4} m A^{2} \omega^{2}$...(i) Average P.E. over one cycle $= \frac{1}{2T} m \omega^{2} A^{2} \int\limits_{0}^{T} \frac{1-\cos 2\omega t}{2} \, dt$$= \frac{1}{4T} m \omega^{2} A^{2} \left[ 1 - \frac{\sin 2\omega t}{2\omega} \right]_{0}^{T}$$= \frac{1}{4T} m \omega^{2} A^{2} [T] = \frac{1}{4} m A^{2} \omega^{2}$...(ii)From (i) and (ii), $E_{k_{av}} = E_{p_{av}}$