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(B) In a simple harmonic motion $(SHM)$ with time period $T$,the displacement is given by $x = A \sin(\omega t)$.The velocity is $v = \frac{dx}{dt} =

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Potential energy

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(B) In a simple harmonic motion $(SHM)$ with time period $T$,the displacement is given by $x = A \sin(\omega t)$.The velocity is $v = \frac{dx}{dt} = A\omega \cos(\omega t)$,which has the same time period $T = 2\, s$.The potential energy is $U = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \sin^2(\omega t) = \frac{1}{4} k A^2 (1 - \cos(2\omega t))$.The kinetic energy is $K = \frac{1}{2} m v^2 = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) = \frac{1}{4} m A^2 \omega^2 (1 + \cos(2\omega t))$.Both potential energy and kinetic energy vary with an angular frequency of $2\omega$,which corresponds to a time period of $T' = \frac{T}{2} = \frac{2\, s}{2} = 1\, s$.Therefore,the potential energy exhibits simple harmonic variation with a period of $1\, s$.

Column-$I$	Column-$II$
$(a)$ $y = \frac{A}{\sqrt{2}}$	$(i)$ $K = \frac{3E}{4}$
$(b)$ $y = \frac{\sqrt{3}A}{2}$	$(ii)$ $K = \frac{E}{4}$
	$(iii)$ $K = \frac{E}{2}$

$A$ vertical mass-spring system executes simple harmonic oscillations with a period of $2\, s$. $A$ quantity of this system which exhibits simple harmonic variation with a period of $1\, s$ is

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