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

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(C) The potential energy is given by $U = 4(1 - \cos 2x)$.The force $F$ acting on the particle is given by $F = -\frac{dU}{dx}$.$F = -\frac{d}{dx} [4(

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$\frac{\pi}{2}$

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(C) The potential energy is given by $U = 4(1 - \cos 2x)$.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 we can approximate $\sin 2x \approx 2x$.Therefore,$F \approx -8(2x) = -16x$.Comparing this with the standard simple harmonic motion equation $F = -kx$,we get the spring constant $k = 16\, N/m$.The mass of the particle is $m = 1\, kg$.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 $T = \frac{2\pi}{\omega} = \frac{2\pi}{4} = \frac{\pi}{2}\, s$.

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 $meters$. The period of small oscillation (in $seconds$) is

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