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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$\mathrm{x}=\mathrm{A} \cos \left(\omega \mathrm{t}+\frac{\pi}{4}\right)$ and $v=\frac{\mathrm{dx}}{\mathrm{dt}}$
$=-A \omega \sin \left(\omega t+\frac{\pi}{4}\right)$
For maximum speed,
$\sin \left(\omega t+\frac{\pi}{4}\right)=1 \Rightarrow \omega t+\frac{\pi}{4}=\frac{\pi}{2}$
or $\omega t=\frac{\pi}{2}-\frac{\pi}{4} \Rightarrow t=\frac{\pi}{4 \omega}$
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