In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency $\omega(t)$ and average amplitude $A(t)$ of the system change with time $t$. Which one of the following options schematically depicts these changes correctly?
Answer: (B) see figure
Concept. $\omega = \sqrt{k/m}$ rises as mass is lost. For a slowly changing frequency $\omega A^2$ stays constant, so $A \propto \omega^{-1/2}$.
Step 1. Frequency. Sand leaks out, $m$ decreases, so $\omega$ increases.
Step 2. Amplitude. The sand leaves with the box's own velocity and exerts no horizontal force; with $\omega A^2$ constant, $A$ decreases as $\omega$ increases. (Each grain carries away its own kinetic energy, so the box's energy $\tfrac12kA^2$ falls.)
Step 3. Choose. $\omega$ rising, $A$ falling: graph (2).
Answer. Graph (2) (Option B).
Solution by Sreeraj P, M.Sc Physics