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(C) For a particle in simple harmonic motion,the total energy $E$ is the sum of kinetic energy $(KE)$ and potential energy $(PE)$.$E = KE + PE$Given t

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$8$

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(C) For a particle in simple harmonic motion,the total energy $E$ is the sum of kinetic energy $(KE)$ and potential energy $(PE)$.$E = KE + PE$Given that $KE = PE$,we have $E = 2 \cdot PE$ or $E = 2 \cdot KE$.The potential energy at displacement $y$ is $PE = \frac{1}{2} k y^2$,and the total energy is $E = \frac{1}{2} k A^2$,where $A$ is the amplitude.Setting $KE = PE$ implies $PE = \frac{1}{2} E$,so $\frac{1}{2} k y^2 = \frac{1}{2} (\frac{1}{2} k A^2)$.This simplifies to $y^2 = \frac{A^2}{2}$,or $y = \frac{A}{\sqrt{2}}$.Using the equation of motion $y = A \sin(\omega t)$,we get $\frac{A}{\sqrt{2}} = A \sin(\omega t)$,which means $\sin(\omega t) = \frac{1}{\sqrt{2}}$.Thus,$\omega t = \frac{\pi}{4}$.Since $\omega = \frac{2\pi}{T}$,we have $(\frac{2\pi}{T}) t = \frac{\pi}{4}$.Solving for $t$,we get $t = \frac{T}{8}$.Comparing this with $\frac{T}{x}$,we find $x = 8$.

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