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Question #42332

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly insulated vertical cylinder of cross-sectional area $A$ contains an ideal gas with adiabatic exponent $\gamma$ . A perfectly insulating, frictionless piston of mass $M$ rests in equilibrium at a height $L$ above the base. The external atmospheric pressure is $P_0$ . The piston is given a small vertical displacement $x$ ( $x \ll L$ ). The total potential energy of the system (gas, piston, and atmosphere) near equilibrium is expressed as $U(x) \approx U_0 + \frac{1}{2} k_{\text{eff}} x^2$ . What is the effective stiffness $k_{\text{eff}}$ of the system?

Options

A.
$\frac{\gamma (P_0 A + M g)}{L}$
B.
$\frac{P_0 A + M g}{L}$
C.
$\frac{\gamma P_0 A + M g}{L}$
D.
$\frac{\gamma P_0 A}{L}$

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

Equilibrium Stability|Energy Partitioning|Adiabatic Process Constraint|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model the stability of a thermodynamic system using the second derivative of the total potential energy.

The primary trap is the distinction between isothermal and adiabatic stiffness.

Students often incorrectly assume an isothermal process, leading to $k_{\text{eff}} = (P_0 A + Mg)/L$ , or they fail to properly apply the polytropic index $\gamma$ when differentiating the pressure-volume relationship $PV^\gamma = \text{constant}$ .

Furthermore, students might forget to include the piston's weight in the equilibrium pressure, leading to an incomplete restoring force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Equilibrium Stability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free