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Question #42517

Difficulty: 8/10 | 🟢 Free

The Question

A metallic block of heat capacity $C$ oscillates horizontally on a frictionless surface, attached to a spring. It passes through a uniform magnetic field, experiencing a damping force due to eddy currents. As a result, its mechanical energy decays as $E(t) = E_0 e^{-t/\tau_1}$ , where $E_0$ is the initial energy. The work done against the damping force is entirely converted into heat within the block. Simultaneously, the block loses heat to the surroundings (at a constant temperature $T_0$ ) according to Newton's law of cooling, with a thermal time constant $\tau_2$ . If $\tau_2 = 2\tau_1$ , and the block is initially at temperature $T_0$ , what is the maximum temperature reached by the block?

Options

A.
$T_0 + \frac{E_0}{2C}$
B.
$T_0 + \frac{E_0}{C}$
C.
$T_0 + \frac{E_0}{4C}$
D.
$T_0 + \frac{2E_0}{3C}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Calorimetry with Heat Loss|Eddy Currents Dissipation|Conservation of Energy

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of mechanical energy dissipation and thermal cooling, a classic trap where students often incorrectly assume that the total dissipated mechanical energy directly determines the maximum temperature (yielding $T_0 + E_0/C$ ).

The trap lies in ignoring that the block continuously loses heat to the environment while it is being heated by eddy currents.

To solve it, students must construct a non-homogeneous first-order differential equation balancing the time-dependent power input ( $P \propto e^{-t/\tau_1}$ ) with the temperature-dependent cooling rate.

Failing to recognize the temperature as a dynamic variable inside the cooling term prevents the correct integration of the energy state.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Calorimetry with Heat Loss
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free