The potential energy of a particle of mass $1\, kg$ in motion along the $x-$ axis is given by $U = 4\,(1 -cos\,2x)$, where $x$ is in $metres$ . The period of small oscillation (in $second$ ) is
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$\mathrm{U}=4(1-\cos 2 \mathrm{x})$
$\because F=-\frac{d U}{d x} \Rightarrow F=-8 \sin 2 x$
For small oscillations, $x$ will be small hence
$F=-8(2 x)=-16 x \quad \Rightarrow k=16$ and $m=1 \mathrm{kg}$
$\therefore \omega^{2}=\frac{\mathrm{k}}{\mathrm{m}}=\frac{16}{1}=16 \Rightarrow \omega=4$
$\Rightarrow \mathrm{T}=\frac{2 \pi}{\omega}=\frac{2 \pi}{4}=\frac{\pi}{2}$
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