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Question #42143

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m$ performs simple harmonic motion with amplitude $A$ and angular frequency $\omega$ . The block acts as a heat source, generating thermal energy at an instantaneous rate $P(t) = \alpha U(t)$ , where $U(t)$ is the instantaneous potential energy of the oscillator (taking $U=0$ at the mean position) and $\alpha$ is a positive constant. This heat is supplied to a mixture of ice and water at $0^{\circ}\mathrm{C}$ . However, due to a thermal gating mechanism, heat is transferred to the mixture only when the block's displacement satisfies $|x| \ge A/2$ . If the latent heat of fusion of ice is $L$ , what is the mass of ice melted in one complete time period of the oscillation?

Options

A.
$\frac{\alpha m \omega A^2}{12 L} (4\pi + 3\sqrt{3})$
B.
$\frac{\pi \alpha m \omega A^2}{3 L}$
C.
$\frac{\pi \alpha m \omega A^2}{2 L}$
D.
$\frac{\alpha m \omega A^2}{12 L} (4\pi - 3\sqrt{3})$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Specific Heat Capacity of Phase Mixtures|Energy Partitioning|Oscillations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to partition energy transfer in an oscillatory system with a position-dependent gating mechanism.

A common trap is assuming that the average power of the oscillator ( $\langle P \rangle = \frac{1}{4}\alpha m \omega^2 A^2$ ) can simply be multiplied by the residence time in the active zone ( $\Delta t = \frac{2}{3}T$ ).

However, because the heat generation is proportional to the potential energy, the power is significantly higher at the extremes of the motion ( $|x| \ge A/2$ ) than near the mean position.

Students must integrate the instantaneous power strictly over the active phase intervals to find the correct thermal energy transferred.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Specific Heat Capacity of Phase Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free