AIIMS 2018 Physics Potential Energy in Oscillations MCQ Question

For a particle oscillating in a potential $V(x) = k|x|^3$, the force can be derived from the potential as $F = -\frac{dV}{dx} = -3k|x|^2 \text{sgn}(x)$. This force leads to non-linear oscillations, and using the method of dimensional analysis, we find that the time period $T$ is proportional to $a^{-1/2}$, where $a$ is the amplitude. Thus, $T \propto \frac{1}{\sqrt{a}}$, confirming option (A).

Option (B) is incorrect because it suggests a direct proportionality to $\sqrt{a}$, which does not align with the derived relationship. Option (C) is incorrect as it implies independence from $a$, which contradicts the dependence established through dimensional analysis.