A system is oscillating with undamped simple harmonic motion. Then the
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Total energy of system is $P E+K E$ remains constant, when system is at mean position, $x=0 \Rightarrow P E=0,$ and $K E=$ max. so average of total energy is equal to total energy at mean position which is maximum kinetic energy.
similarly, at extreme points, $K E=0$ and $P E=$ $max.$
so average of total energy is equal to total energy at extreme position which is maximum potential energy.
Now, if $x=A \sin \omega t \Rightarrow v=A \omega \cos \omega t$
Maximum velocity is $v_{\max }=A \omega$ and $\mathrm{rms}$ value of velocity is $A \omega / \sqrt{2}=v_{\max } / \sqrt{2}(\because \mathrm{rms}$ value of $\cos \theta$ is $1 / \sqrt{2}$ )
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