STANDARD Physics Simple Harmonic Motion MCQ Question

In simple harmonic motion (SHM), the relationship between acceleration $a$, velocity $v$, and displacement $x$ is governed by the equations of motion, where $a = -\omega^2 x$ and $v = \omega A \sin(\omega t + \phi)$, with $\omega = \frac{2\pi}{T}$. Option D, $a^2 T + 4\pi^2 v^2$, simplifies to a constant value because $a^2$ and $v^2$ are both proportional to $A^2$ and do not change with time, thus making the entire expression time-independent.

In contrast, options A, B, and C involve terms that vary with time due to the periodic nature of SHM, as $a$ and $v$ fluctuate with the displacement $x$ over each cycle, leading to changes in the ratios and sums presented in those options.