A particle is executing linear SHM. The average kinetic energy and average potential energy, over a period of oscillation, respectively are ${\mathrm{K}}_{\mathrm{av}} \mathrm{and} {\mathrm{U}}_{\mathrm{av}}$. Then,

$\mathrm{K} = \frac{1}{2}{\mathrm{mv}}^2 = \frac{1}{2}{\mathrm{mA}}^2{\mathrm{\omega }}^2{\cos }^2\mathrm{\omega t}$

Average value of ${\cos }^2 \mathrm{\omega t} \mathrm{is} \frac{1}{2} \mathrm{over} \mathrm{one} \mathrm{cycle}$

$\therefore {\mathrm{K}}_{\mathrm{av}} = \frac{1}{4}{\mathrm{m\omega }}^2{\mathrm{A}}^2$------------(i)

      $\mathrm{U} = \frac{1}{2}{\mathrm{m\omega }}^2{\mathrm{x}}^2 = \frac{1}{2}{\mathrm{m\omega }}^2{\mathrm{A}}^2{\sin }^2\mathrm{\omega t}$

Average value of ${\sin }^2 \mathrm{\omega t} \mathrm{is} \frac{1}{2} \mathrm{over} \mathrm{one} \mathrm{cycle}.$

$\therefore {\mathrm{U}}_{\mathrm{av}} = \frac{1}{4}{\mathrm{m\omega }}^2{\mathrm{A}}^2$---------(iii)

From Eqs. (i) and (ii), we get ${\mathrm{K}}_{\mathrm{av}} = {\mathrm{U}}_{\mathrm{av}}$