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