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 where a box filled with sand is attached to the spring, as sand leaks from the box, it results in a change in both the mass of the box and the spring dynamics. Let's analyze how this affects the average frequency (ω(t)) and amplitude (A(t)) of the oscillating system over time:

Frequency Analysis:

The natural frequency of a simple harmonic oscillator is given by:

ω = √(k/m)

where k is the spring constant and m is the mass of the system. As sand leaks out, the mass m decreases, leading to an increase in the frequency ω because the mass (m) is in the denominator. Thus, ω(t) increases over time.

Amplitude Analysis:

The amplitude of oscillation in a damped system can be influenced by external factors such as the loss of mass. When sand leaks out, the energy dissipation remains similar, but the effective damping due to air resistance relative to the mass increases, leading to a decrease in amplitude over time due to increased damping relative to reduced mass.

Therefore, A(t) decreases over time.

Conclusion:

The correct schematic for these changes would show an increase in frequency (ω) with time and a decrease in amplitude (A) with time. This is best represented by Figure 1, as it accurately depicts the described changes: increasing frequency and decreasing amplitude over time.