A particle of mass $m$ executes simple harmonic motion with amplitude $a$ and frequency $v$. The average kinetic energy during its motion from the position of equilibrium to the end is
AIEEE 2007, Medium
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$The\, instantaneous \,kinetic\, energy\, of$
$\,a\, particle\, executing \,S.H.M.\, is \,given \,by$
$ 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}$
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