A particle performs $SHM$ with a period $T$ and amplitude $a.$ The mean velocity of the particle over the time interval during which it travels a distance $a/2$ from the extreme position is
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$x=A \cos \omega t \Rightarrow \frac{A}{2}=A \cos \omega t$
$\Rightarrow \quad \cos \omega t=\frac{1}{2}=\cos \frac{\pi}{3}$
$\frac{2 \pi}{T} t=\frac{\pi}{3}$
$\Rightarrow \quad t=\frac{T}{6}$
$v_{\mathrm{av}}=\frac{a / 2}{t}=\frac{a / 2}{T / 6}=\frac{3 a}{T}$
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