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Question #43920

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The Question

A horizontal cylinder of cross-sectional area $A$ contains an ideal monoatomic gas at an initial equilibrium pressure $P_0$ and volume $V_0$ . A movable piston of mass $m$ seals the gas and is connected to a fixed vertical wall by a linear spring of spring constant $k$ . The region on the other side of the piston is a vacuum. At equilibrium, the spring is in a stretched state. An external agent slowly displaces the piston outward by a small distance $x$ (where $x \ll V_0/A$ ). Let $W_a$ be the total work done by the external agent if the gas undergoes a reversible adiabatic process, and $W_i$ be the total work done if the gas undergoes a reversible isothermal process. Which of the following expressions correctly represents the difference $(W_a - W_i)$ to the leading order in $x$ ?

Options

A.
$\frac{P_0 A^2 x^2}{3 V_0}$
B.
$\frac{2 P_0 A^2 x^2}{3 V_0}$
C.
$\frac{5 P_0 A^2 x^2}{6 V_0}$
D.
$0$

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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

Effective Spring Constant|Adiabatic/Isothermal Processes via KTG|Integration

Why This Question Is Tricky

Common Mistakes

Students often misapply the work formula by only considering the first-order approximation $W \approx P \Delta V = P_0 A x$ , leading them to incorrectly conclude that the work done is identical for both processes (Option D).

Furthermore, they fail to recognize that the gas acts as a nonlinear spring whose 'effective spring constant' is derived from the second derivative of its potential energy function $d^2U/dx^2$ .

By equating the external work to the change in total potential energy, the first-order terms perfectly cancel due to the equilibrium condition ( $k x_0 = P_0 A$ ).

The difference in work strictly emerges from the second-order Taylor expansion terms, which depend on the polytropic index ( $\gamma$ vs 1) governing the curvature of the potential well.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Effective Spring Constant
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free