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An object of mass $4 \mathrm{~kg}$ is moving along x-axis and its $\mathrm{P E}$ as a function of $\mathrm{x}$ varies as $\mathrm{U(x)=4(1-\cos 2 x) J}$ , the time period of small oscillations is-

Option: 1
$\pi\: \mathrm{sec}$

Option: 2
$2 \pi \: \mathrm{sec}$

Option: 3
$4 \pi\: \mathrm{sec}$

Option: 4
$2 \: \mathrm{sec}$

Answers (1)

best_answer

$\mathrm{F=-\frac{d U}{d x}=-\frac{d}{d x}[4(1-\cos 2 x)]}$

$\mathrm{ f=-8 \sin 2 x }$ ________(1)

For small $\mathrm{ x }$

$\mathrm{ \sin 2 x=2 x }$

$\mathrm{ f=-8 \times 2 x }$

$\mathrm{ or \: \: a=\frac{f}{m}=-\frac{8 \times 2 x}{4}=-4 x }$

on comparing with $\mathrm{ a=-\omega^2 x }$ , we have

$\mathrm{ \omega=\sqrt{4}\: \mathrm{rad} / \mathrm{s} \quad \& \quad T=\frac{2 \pi}{\omega} }$

$\mathrm{ \omega=2 \: \mathrm{rad} / \mathrm{s} \quad T=\frac{2 \pi}{2} }$

$\mathrm{ T=\pi\: \mathrm{sec} }$

Hence option 1 is correct.

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