First slide

Simple hormonic motion

Question

A simple pendulum performs simple harmonic motion about.x = 0 with an amplitude a and time period T. The speed of the pendulum at $x = \frac{a}{2}$ will be

Moderate

A

$\frac{πa \sqrt{3}}{2 T}$
B

$\frac{πa}{T}$
C

$\frac{3 π^{2} a}{T}$
D

$\frac{πa \sqrt{3}}{T}$

Solution

$v = \frac{dy}{dt} = Aω \cos ωt = Aω \sqrt{1 - \sin^{2} ωt}$
= $ω \sqrt{A^{2} - y^{2}}$
Here, y = $\frac{a}{2}$
$∴ v = ω \sqrt{a^{2} - \frac{a^{2}}{4}} = ω \sqrt{\frac{3 a^{2}}{4}} = \frac{2 π}{T} \frac{a \sqrt{3}}{2} = \frac{πa \sqrt{3}}{T}$

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