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(D) The displacement of a particle in simple harmonic motion $(SHM)$ is given by $x = a \sin(\omega t)$.The velocity of the particle is given by $v =

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$\frac{\sqrt{3} \pi a}{T}$

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(D) The displacement of a particle in simple harmonic motion $(SHM)$ is given by $x = a \sin(\omega t)$.The velocity of the particle is given by $v = \omega \sqrt{a^2 - x^2}$.Given the periodic time is $T$,the angular frequency is $\omega = \frac{2\pi}{T}$.At $x = \frac{a}{2}$,the velocity is:$v = \frac{2\pi}{T} \sqrt{a^2 - (\frac{a}{2})^2}$$v = \frac{2\pi}{T} \sqrt{a^2 - \frac{a^2}{4}}$$v = \frac{2\pi}{T} \sqrt{\frac{3a^2}{4}}$$v = \frac{2\pi}{T} \cdot \frac{\sqrt{3}a}{2}$$v = \frac{\sqrt{3} \pi a}{T}$.

$A$ simple pendulum starts oscillating simple harmonically from its mean position $(x=0)$ with amplitude '$a$' and periodic time '$T$'. The magnitude of velocity of the pendulum at $x=\frac{a}{2}$ is

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