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Question #42544

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated cylinder contains an ideal monoatomic gas at initial pressure $P_0$ and volume $V_0$ . A piston of cross-sectional area $A$ is connected to a rack-and-pinion mechanism. The pinion gear has radius $R$ and is hinged inside the cylinder such that any heat generated by friction at the hinge is entirely transferred to the gas. The hinge provides a constant frictional torque $\tau$ opposing any rotation. An external agent slowly rotates the gear, compressing the gas to a final volume $\frac{V_0}{8}$ . The linear displacement $x$ of the piston is related to the angular displacement $\theta$ of the gear by $x = R\theta$ . Which of the following expressions gives the final pressure of the gas?

Options

A.
$32 P_0 + \frac{62}{5} \frac{\tau}{A R}$
B.
$32 P_0 + \frac{14}{3} \frac{\tau}{A R}$
C.
$32 P_0 - \frac{62}{5} \frac{\tau}{A R}$
D.
$32 P_0 + \frac{64}{5} \frac{\tau}{A R}$

Check Your Answer

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

First Law Application|Rigid Body Constraints|Molar Heat Capacity (Cv vs Cp)

Why This Question Is Tricky

Common Mistakes

This question tests the candidate's ability to couple non-conservative mechanical work with thermodynamic state equations.

The primary trap lies in the bifurcation of work: students often incorrectly assume that the external work done on the system solely contributes to boundary work ( $-PdV$ ), ignoring that the frictional dissipation at the hinge acts as a continuous internal heat source ( $dQ > 0$ ).

A common mistake (leading to Option B) is to treat the process as purely adiabatic ( $PV^{5/3} = \text{const}$ ) and erroneously add the total frictional heat at the end of the process at constant volume.

The correct approach requires formulating a modified differential equation from the First Law where $dQ$ is explicitly a continuous function of $dV$ , leading to a shifted polytropic constant.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	First Law Application
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free