Question #81512

Assertion (A): A bucket containing sand of total instantaneous mass $M(t)$ is suspended from a vertical spring of stiffness $k$ and oscillates. Sand leaks out from a hole at the bottom at a constant, very slow rate $\mu = -\frac{dM}{dt}$ (with zero relative velocity with respect to the bucket). During this process, the amplitude of oscillation $A$ of the bucket decreases as $A \propto (M(t))^{1/4}$.

Reason (R): The leaking sand exerts zero thrust force, yielding the equation of motion $M(t)\ddot{z} + kz = 0$ about the instantaneous mean position $z=0$. The continuous shedding of mass removes kinetic energy from the system at a rate $\frac{1}{2}\dot{M}\dot{z}^2$, causing the amplitude to decrease, whereas applying the standard adiabatic invariant $\frac{E}{\omega} = \text{constant}$ would incorrectly predict an increasing amplitude.