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 ω(t) and average amplitude A(t) of the system change with time t. Which one of the following options schematically depicts these changes correctly? (a) (b) (c) (d)

Answer: (d). Sand leaks out, so the mass m decreases and ω = √(k/m) increases with time. Sand leaving vertically takes its own share of energy; for the box, x + ω²(t) x = 0 with slowly changing ω. For such slow change ω A² stays constant (adiabatic invariant), so A ∝ (1)/(√(ω)) decreases. Graph with ω rising and A falling: the answer is (d).

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Spring Systems

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?

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Answer: (d)

Sand leaks out, so the mass $m$ decreases and $\omega = \sqrt{\frac{k}{m}}$ increases with time.
Sand leaving vertically takes its own share of energy; for the box, $\ddot{x} + \omega^2(t)\,x = 0$ with slowly changing $\omega$ .
For such slow change $\omega A^2$ stays constant (adiabatic invariant), so $A \propto \frac{1}{\sqrt{\omega}}$ decreases.
Graph with $\omega$ rising and $A$ falling: the answer is (d).

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