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$(B)$ The displacement of a particle executing $SHM$ is given by $y = a \sin(\omega t)$.The velocity of the particle is $u = \frac{dy}{dt} = a\omega \

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changes periodically with a frequency of $2v$

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$(B)$ The displacement of a particle executing $SHM$ is given by $y = a \sin(\omega t)$.The velocity of the particle is $u = \frac{dy}{dt} = a\omega \cos(\omega t)$.The kinetic energy $K$ is given by $K = \frac{1}{2}mu^2 = \frac{1}{2}m(a\omega \cos(\omega t))^2 = \frac{1}{2}m\omega^2 a^2 \cos^2(\omega t)$.Using the trigonometric identity $\cos^2(	heta) = \frac{1 + \cos(2	heta)}{2}$, we get $K = \frac{1}{4}m\omega^2 a^2 (1 + \cos(2\omega t))$.Since the frequency of the $SHM$ is $v = \frac{\omega}{2\pi}$, the frequency of the kinetic energy oscillation is determined by the term $\cos(2\omega t)$, which is $v' = \frac{2\omega}{2\pi} = 2v$.Therefore, the kinetic energy changes periodically with a frequency of $2v$.

When a particle executing $SHM$ oscillates with a frequency $v$, then the kinetic energy of the particle

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