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(A) The potential energy $(P.E.)$ of a particle in $S.H.M.$ is given by $U = \frac{1}{2} k x^2$.Substituting $x = A \cos \omega t$,we get $U = \frac{1

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$I, III$

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(A) The potential energy $(P.E.)$ of a particle in $S.H.M.$ is given by $U = \frac{1}{2} k x^2$.Substituting $x = A \cos \omega t$,we get $U = \frac{1}{2} k A^2 \cos^2 \omega t = \frac{1}{2} k A^2 \left( \frac{1 + \cos 2 \omega t}{2} \right)$.At $t = 0$,$x = A$,so $U$ is maximum. Graph $I$ shows $P.E.$ vs $t$ starting from a maximum value at $t = 0$,which matches the equation $U \propto \cos^2 \omega t$.As a function of displacement $x$,$U = \frac{1}{2} k x^2$,which is a parabola opening upwards with its minimum at $x = 0$. Graph $III$ represents this parabolic variation of $P.E.$ with respect to $x$.Thus,graphs $I$ and $III$ are correct.

For a particle executing $S.H.M.$,the displacement $x$ is given by $x = A \cos \omega t$. Identify the graphs which represent the variation of potential energy $(P.E.)$ as a function of time $t$ and displacement $x$.

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