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A simple pendulum of length $1\, m$ is oscillating with an angular frequency $10\, rad/s$. The support of the pendulum starts oscillating up and down with a small angular frequency of $1\, rad/s$ and an amplitude of $10^{-2}\, m$. The relative change in the angular frequency of the pendulum is best given by
$10^{-3} rad/s$
$1\,rad/s$
$10^{-1} rad/s$
$10^{-5} rad/s$
Solution
Angular frequency of pendulum
$\omega \propto \sqrt{\frac{g_{e f f}}{\ell}}$
$\therefore \quad \frac{\Delta \omega}{\omega}=\frac{1}{2} \frac{\Delta g_{e f f}}{g_{e f f}}$
$\Delta \omega=\frac{1}{2} \frac{\Delta g}{g} \times \omega$
$[ \omega_{s}=$ angular frequency of support]
$\frac{\Delta \omega}{\omega}=\frac{1}{2} \times \frac{\Delta g}{g}$
$=\frac{1}{2} \times \frac{2\left(\mathrm{A} \omega_{5}^{5}\right)}{10}$
$\Rightarrow \frac{\Delta \omega}{\omega}=\frac{1 \times 10^{-2}}{10}=10^{-3}$
