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(C) The velocity $v$ of a particle in simple harmonic motion is given by the formula $v = \omega \sqrt{a^2 - x^2}$, where $\omega$ is the angular freq

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$\frac{4\sqrt{2}\pi a}{3T}$

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(C) The velocity $v$ of a particle in simple harmonic motion is given by the formula $v = \omega \sqrt{a^2 - x^2}$, where $\omega$ is the angular frequency and $a$ is the amplitude.Given that the periodic time is $T$, the angular frequency is $\omega = \frac{2\pi}{T}$.We need to find the speed at $x = \frac{a}{3}$.Substituting the values into the formula:$v = \left( \frac{2\pi}{T} \right) \sqrt{a^2 - \left( \frac{a}{3} \right)^2}$$v = \left( \frac{2\pi}{T} \right) \sqrt{a^2 - \frac{a^2}{9}}$$v = \left( \frac{2\pi}{T} \right) \sqrt{\frac{8a^2}{9}}$$v = \left( \frac{2\pi}{T} \right) \left( \frac{2\sqrt{2}a}{3} \right)$$v = \frac{4\sqrt{2}\pi a}{3T}$

$A$ particle is performing simple harmonic motion about $x = 0$ with an amplitude $a$ and periodic time $T$. The speed of the particle at $x = \frac{a}{3}$ will be

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