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Question #43929

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal cylindrical tube of cross-sectional area $A = 20 \text{ cm}^2$ and length $L = 0.7 \text{ m}$ is closed at one end. The other end is fitted with a movable piston of mass $M = 90 \text{ g}$ . The tube contains an ideal gas of total mass $m_{gas} = 30 \text{ g}$ at an equilibrium pressure $P_0 = 10^5 \text{ Pa}$ . The gas undergoes adiabatic processes with an adiabatic index $\gamma = 1.4$ . The piston is connected to a fixed support via a linear spring of stiffness $k = 100 \text{ N/m}$ , which is relaxed when the piston is in equilibrium. The piston experiences a constant kinetic friction force $F_k = 5 \text{ N}$ while moving. The piston is displaced outwards by a distance $x_0 = 4 \text{ cm}$ from its equilibrium position and released from rest. Assume the velocity of the gas particles varies linearly from zero at the fixed closed end to the velocity of the piston at the moving end. What is the maximum speed of the piston during its subsequent motion?

Options

A.
$\frac{3}{\sqrt{2}} \text{ m/s}$
B.
$\sqrt{5} \text{ m/s}$
C.
$2\sqrt{2} \text{ m/s}$
D.
$\frac{\sqrt{15}}{2} \text{ m/s}$

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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

End Correction|Effective Spring Constant|Kinetic Friction Transition|Integration

Why This Question Is Tricky

Common Mistakes

This problem tests the coupling of acoustic/gas impedance with mechanical elements under non-conservative forces.

A common trap is failing to recognize that the trapped gas acts simultaneously as an adiabatic spring ( $k_{gas} = \gamma P_0 A / L$ ) and a distributed inertial mass.

Ignoring the effective mass of the gas ( $m_{gas}/3$ ) leads to Option B.

Ignoring the kinetic friction's shift of the equilibrium position (or ignoring friction entirely in the energy equation) leads to Option C.

Using the full gas mass instead of integrating the velocity profile (Rayleigh's method) leads to Option D.

The transition from static to kinetic friction creates a piecewise equation of motion, shifting the point of maximum velocity away from the natural equilibrium $x=0$ to $x = F_k/K_{eff}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	End Correction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free