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Question #42518

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylindrical rod of length $L$ , cross-sectional area $A$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ is rigidly clamped between two immovable walls at an ambient temperature $T_0$ . It is heated via a calorimetry setup involving a hot liquid, reaching an initial uniform temperature $T_0 + \Delta T_0$ . The combined heat capacity of the rod (excluding elastic energy) and the liquid is $C_0$ . The system is then allowed to cool. The calorimeter loses heat to the surroundings at a rate $\frac{dQ}{dt} = -k(T - T_0)$ , where $k$ is a positive constant and $T$ is the instantaneous temperature. Assuming the rod and liquid are always in thermal equilibrium, what is the time $t$ taken for the system to cool to $T_0 + \frac{\Delta T_0}{2}$ ?

Options

A.
$\frac{C_0}{k} \ln 2 + \frac{Y A L \alpha^2 \Delta T_0}{2k}$
B.
$\frac{C_0}{k} \ln 2 + \frac{Y A L \alpha^2 \Delta T_0}{k}$
C.
$\frac{C_0}{k} \ln 2 - \frac{Y A L \alpha^2 \Delta T_0}{2k}$
D.
$\frac{C_0}{k} \ln 2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Calorimetry with Heat Loss|Thermal Stress Calculation|Internal Energy Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of macroscopic thermodynamics with solid mechanics.

The primary trap is neglecting the elastic potential energy stored in the rod due to thermal stress when clamped between rigid walls.

Students commonly assume the heat capacity of the system is constant ( $C_0$ ) and use $dQ = C_0 dT$ , leading to the standard exponential cooling time (Option D).

However, the constrained expansion results in internal elastic potential energy being stored in the rod, $U_{elastic} = \frac{1}{2} Y A L \alpha^2 (\Delta T)^2$ .

This makes the effective heat capacity temperature-dependent ( $C_{eff} = C_0 + Y A L \alpha^2 \Delta T$ ), requiring integration of a modified differential equation to find the correct cooling time.

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