The instantaneous displacement of a simple pe | vedclass

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Question

(A) The displacement is given by $x = A \cos \left( \omega t + \frac{\pi}{4} \right)$.The velocity $v$ is the derivative of displacement with respect

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$\frac{\pi}{4\omega}$

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(A) The displacement is given by $x = A \cos \left( \omega t + \frac{\pi}{4} \right)$.The velocity $v$ is the derivative of displacement with respect to time:$v = \frac{dx}{dt} = -A \omega \sin \left( \omega t + \frac{\pi}{4} \right)$.For the speed to be maximum,the magnitude of the sine function must be maximum,i.e.,$|\sin \left( \omega t + \frac{\pi}{4} \right)| = 1$.This occurs when $\omega t + \frac{\pi}{4} = \frac{\pi}{2}$ (considering the first positive time value).Solving for $t$:$\omega t = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}$.Therefore,$t = \frac{\pi}{4\omega}$.

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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