Question

In an oscillating spring mass system, a spring 9 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?

• A. Option 1
• B. Option 2
• C. Option 3
• D. Option 4

Answer 1

Answer: (C)

3

Answer 2

The angular frequency of a spring-mass system is $\omega = \sqrt{k/m}$. As sand leaks, the mass $m(t)$ decreases, causing the frequency $\omega(t)$ to increase with time.

For slowly varying mass, the adiabatic invariant $E/\omega$ is approximately constant. The energy of the oscillator is $E = \frac{1}{2}kA^2$. Thus, $\frac{\frac{1}{2}kA^2}{\sqrt{k/m}} = constant$, which implies $A^2 \sqrt{m} = constant$, or $A \propto m^{-1/4}$. Since the mass $m(t)$ is decreasing with time, the amplitude $A(t)$ increases with time.

Therefore, both the average frequency and the average amplitude of the system increase with time.