A simple perdulum performs simple harmonic motion about $x=0$ with an amplitude $a$ and time period $T$. The speed of the pendulum at $x=a/2$ will be
AIPMT 2009, Medium
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For simple harmonic motion,
$v=\omega \sqrt{a^{2}-x^{2}}$
When $x=\frac{a}{2}, v=\omega \sqrt{a^{2}-\frac{a^{2}}{4}}=\omega \sqrt{\frac{3}{4} a^{2}}$
As $\omega=\frac{2 \pi}{T}, \quad \therefore \quad v=\frac{2 \pi}{T} \cdot \frac{\sqrt{3}}{2} a \Rightarrow \quad v=\frac{\pi \sqrt{3} a}{T}$
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