The instantaneous displacement of a simple pendulum oscillator is given by $x = A\cos \left( {\omega t + \frac{\pi }{4}} \right)$. Its speed will be maximum at time
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(a) $x = A\cos \left( {\omega t + \frac{\pi }{4}} \right)$ and
$v = \frac{{dx}}{{dt}} = - A\omega \sin \left( {\omega \,t + \frac{\pi }{4}} \right)$
For maximum speed, $\sin \,\left( {\omega \,t + \frac{\pi }{4}} \right) = 1$
==> $\omega \,t + \frac{\pi }{4} = \frac{\pi }{2}$ or $\omega \,t = \frac{\pi }{2} - \frac{\pi }{4}$
==> $t = \frac{\pi }{{4\omega }}$
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