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(C) The period of oscillation $T$ is given by $T = 2\pi \sqrt{\frac{m}{k}}$.Given $T = \frac{\pi}{2}$ and $m = 1\,kg$,we have $\frac{\pi}{2} = 2\pi \s

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$-16\,\cos\,x$

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(C) The period of oscillation $T$ is given by $T = 2\pi \sqrt{\frac{m}{k}}$.Given $T = \frac{\pi}{2}$ and $m = 1\,kg$,we have $\frac{\pi}{2} = 2\pi \sqrt{\frac{1}{k}}$.Squaring both sides: $\frac{1}{4} = 4 \cdot \frac{1}{k}$,which gives $k = 16$.For small oscillations,the restoring force is $F = -kx = -16x$.Since $\sin x \approx x$ for small $x$,we can write $F = -16 \sin x$.The potential energy $U$ is given by $U = -\int F dx = -\int (-16 \sin x) dx = -16 \cos x + C$.Thus,the potential energy may be $-16 \cos x$.

The mass of a particle is $1\,kg$ and it is moving along the $x-$axis. The period of its small oscillation is $\frac{\pi}{2}$. Its potential energy may be:

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