\(\,a\, particle\, executing \,S.H.M.\, is \,given \,by\)

\( K=\frac{1}{2} m a^{2} \omega^{2} \sin ^{2} \omega t\)

\(\therefore \,average\,K.E =  < K >  =  < \frac{1}{2}m{\omega ^2}{a^2}{\sin ^2}\omega t > \)

\(= \frac{1}{2} m \omega^{2} a^{2}<\sin ^{2} \omega t>\)

\(= \frac{1}{2} m \omega^{2} a^{2}\left(\frac{1}{2}\right) \quad\left(\because<\sin ^{2} \theta>=\frac{1}{2}\right)\)

\(= \frac{1}{4} m \omega^{2} a^{2}=\frac{1}{4} m a^{2}(2 \pi v)^{2}(\because \omega=2 \pi v)\)

\(   \text { or, }  < K >=\pi^{2} m a^{2} v^{2}\)