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Question #42299

Difficulty: 8/10 | 🟢 Free

The Question

A rigid diathermic container of negligible mass and internal volume $V$ rests on a frictionless horizontal surface. It initially contains an ideal gas of mass $m_0$ and molar mass $\mu$ at temperature $T$ . A small aperture of area $A$ is opened on one side, allowing the gas to effuse into a vacuum. The gas temperature is maintained constant at $T$ by the surroundings. The effusing gas exerts a recoil thrust $F = PA$ , where $P$ is the instantaneous internal pressure. Using the kinetic theory of gases for the effusion rate, determine the displacement of the container by the time the mass of the gas inside reduces to $\frac{m_0}{2}$ .

Options

A.
$\frac{\pi V}{A} (\ln 2)^2$
B.
$\frac{\pi V}{4 A}$
C.
$\frac{\pi V}{2 A} \ln 2$
D.
$\frac{2 \pi V}{A} (\ln 2)^2$

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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

Recoil and Thrust Forces|Effusion and Diffusion|Work-Energy Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the mechanics of variable-mass systems coupled with the kinetic theory of gases.

A common trap is assuming the recoil force is constant or incorrectly applying the work-energy theorem without accounting for the kinetic energy carried away by the effusing gas.

Students must recognize that while the thrust force $F(t)$ and mass $m(t)$ are time-dependent, the effective exhaust velocity $v_e = F / |dm/dt|$ remains constant in an isothermal process.

Setting up the differential equation $dx = v(m) dt$ and transforming the integration variable from time to mass is crucial to finding the exact displacement.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Recoil and Thrust Forces
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free