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(A) The displacement of the particle is given by $x = A \cos \omega t$.Potential energy $(P.E.)$ is given by $U = \frac{1}{2} k x^2$.Substituting $x$,

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$I, III$

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(A) The displacement of the particle is given by $x = A \cos \omega t$.Potential energy $(P.E.)$ is given by $U = \frac{1}{2} k x^2$.Substituting $x$,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 $P.E. = \frac{1}{2} k A^2$ (maximum).Graph $I$ shows $P.E.$ vs $t$ starting from a maximum value at $t = 0$,which matches the equation.As a function of displacement $x$,$U = \frac{1}{2} k x^2$,which is a parabola opening upwards symmetric about the $P.E.$ axis.Graph $III$ shows $P.E.$ vs $x$ as a parabola symmetric about the $P.E.$ axis,which matches the equation.Thus,graphs $I$ and $III$ represent the variation of $P.E.$ with $t$ and $x$ respectively.

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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