The potential energy of a particle of mass 1, | vedclass

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(C) Given mass $m = 1\,kg$ and potential energy $U = 4(1 - \cos 2x)\,J$.The force $F$ acting on the particle is given by $F = -\frac{dU}{dx}$.$F = -\f

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$\frac{\pi}{2}$

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(C) Given mass $m = 1\,kg$ and potential energy $U = 4(1 - \cos 2x)\,J$.The force $F$ acting on the particle is given by $F = -\frac{dU}{dx}$.$F = -\frac{d}{dx} [4(1 - \cos 2x)] = -4(0 - (-\sin 2x) \cdot 2) = -8 \sin 2x$.For small oscillations,$x$ is very small,so $\sin 2x \approx 2x$.Thus,$F \approx -8(2x) = -16x$.Comparing this with the standard $SHM$ equation $F = -kx$,we get the force constant $k = 16\,N/m$.The angular frequency $\omega$ is given by $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{16}{1}} = 4\,rad/s$.The time period $T$ is given by $T = \frac{2\pi}{\omega} = \frac{2\pi}{4} = \frac{\pi}{2}\,sec$.

The potential energy of a particle of mass $1\,kg$ in motion along the $x-$axis is given by $U = 4(1 - \cos 2x)\,J,$ where $x$ is in $meters$. The period of small oscillations (in $sec$) is

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