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Question #84402

Difficulty: 5/10 | 🟢 Free

The Question

A vertical, thermally conducting cylinder of cross-sectional area $A$ is closed at the bottom and fitted with a frictionless, airtight piston of mass $M_p$ . The cylinder contains an ideal gas of total mass $M_g$ and molar mass $\mu$ . The entire system is maintained at a constant temperature $T$ . Due to the large height of the gas column, the hydrostatic pressure gradient within the gas cannot be ignored. The piston is slightly displaced downwards from its equilibrium position and released. Assuming the gas continuously maintains hydrostatic equilibrium and its kinetic energy is negligible during the motion, what is the angular frequency $\omega$ of the resulting vertical oscillations?

(Given: Acceleration due to gravity is $g$ , Universal gas constant is $R$ , and the space above the piston is a vacuum)

Options

A.
$\sqrt{\frac{\mu g^2}{R T} \left( \frac{M_p + M_g}{M_g} \right)}$
B.
$\sqrt{\frac{\mu g^2}{R T} \left( \frac{M_p}{M_g} \right)}$
C.
$\sqrt{\frac{\mu g^2}{R T} \left( \frac{M_p}{M_p + M_g} \right)}$
D.
$\sqrt{\frac{\mu g^2}{R T} \left( \frac{M_g}{M_p} \right)}$

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

Ideal Gas Assumptions|Vertical Oscillations|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to account for the gravitational potential energy of a gas column during vertical oscillations.

The primary trap lies in misapplying the ideal gas law by assuming a uniform pressure distribution ( $P = M_p g / A$ ) and ignoring the work done against the gas's own weight.

Students typically calculate the spring constant as $k = P A / h$ , which erroneously treats the gas as a massless spring.

By forcing the inclusion of the gas mass ( $M_g$ ), the effective spring constant shifts to a coupled relation involving both the piston and the gas mass.

This requires integrating the hydrostatic pressure gradient to find the true restoring force, reflecting the change in the center of mass of the gas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Ideal Gas Assumptions
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free