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? 

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

Hint:

Frequency of a spring-mass system is inversely proportional to the square root of the mass. Amplitude decreases due to energy loss from damping or mass leaving the system.

Answer 1

The Correct Option is A

In a spring-mass system with a sand-filled box, sand leakage alters both the system's mass and spring dynamics. This analysis examines the resulting changes in average frequency (ω(t)) and amplitude (A(t)) over time:

Frequency Analysis:

The natural frequency of a simple harmonic oscillator is calculated as:

\(\omega = \sqrt{k/m}\)

where k denotes the spring constant and m represents the system's mass. As sand leaks out, m decreases, inversely increasing ω because mass (m) is the denominator. Consequently, ω(t) rises over time.

Amplitude Analysis:

Oscillation amplitude in a damped system can be affected by factors like mass loss. Sand leakage maintains similar energy dissipation, but air resistance's damping effect relative to the decreasing mass intensifies. This leads to a decline in amplitude over time due to enhanced relative damping.

Therefore, A(t) diminishes over time.

Conclusion:

The appropriate schematic illustrating these changes would depict frequency (ω) increasing with time and amplitude (A) decreasing with time. Figure 1 best represents these observed trends: escalating frequency and reducing amplitude over the duration.