In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly o…

Question

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?

1. A. ω(t) increases, A(t) stays constant
2. B. ω(t) increases, A(t) decreases (Correct answer)
3. C. ω(t) increases, A(t) increases
4. D. ω(t) decreases, A(t) decreases

Correct Answer

Option B — ω(t) increases, A(t) decreases

Detailed Solution & Explanation

The correct answer is $\omega(t)$ increases and $A(t)$ decreases (Option 2 in the paper).

Key Points

• Time period $T=2\pi\sqrt{\dfrac{m}{k}}$. As sand leaks out, $m$ decreases, so $T$ decreases and $\omega=\dfrac{2\pi}{T}$ increases.
• At equilibrium the spring stretch is $x_0=\dfrac{mg}{k}$. As $m$ decreases, the equilibrium length decreases, so the amplitude measured about it also decreases.
• Hence $\omega(t)$ rises while $A(t)$ falls with time.