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Question #42539

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated horizontal cylinder of cross-sectional area $A$ contains an ideal rigid diatomic gas. A frictionless, insulating piston of mass $M$ confines the gas to a column of length $x$ . The fixed end of the cylinder and the piston are coated with conducting plates, forming an isolated parallel plate capacitor with a constant charge $Q$ . Due to the gas density shifts, its dielectric constant varies with the column length as $K = \frac{L_0}{x}$ (assume $x \ll L_0$ ). If the piston is slightly displaced from its equilibrium position, what is the angular frequency $\omega$ of the resulting adiabatic oscillations?

Options

A.
$Q \sqrt{\frac{12}{5 L_0 \epsilon_0 A M}}$
B.
$Q \sqrt{\frac{6}{5 L_0 \epsilon_0 A M}}$
C.
$Q \sqrt{\frac{7}{5 L_0 \epsilon_0 A M}}$
D.
$Q \sqrt{\frac{10}{5 L_0 \epsilon_0 A M}}$

Check Your Answer

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

Internal Energy Dependency|Dielectric Constant (K) Shift|Conservative Field Path Independence

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of thermodynamics and electrostatics, specifically focusing on the work-energy coupling when a dielectric depends on the mechanical state (volume/density).

The lethal trap occurs when students calculate the electrostatic force using the standard formula $F = \frac{Q^2}{2 \epsilon_0 A K}$ without realizing that $K$ itself is a function of $x$ .

By failing to differentiate $K(x)$ via $F = -\nabla U_E$ , they miss a factor of 2 in the electrostatic force and stiffness (leading to Option B).

Furthermore, they must correctly apply equipartition for a diatomic gas ( $\gamma = 7/5$ ) and recognize that the changing electrostatic field acts as an active 'spring' ( $k_e \neq 0$ ), adding to the gas stiffness.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free