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Question #42481

Difficulty: 8/10 | 🟢 Free

The Question

A vertical, thermally insulated cylinder contains an ideal monatomic gas. A frictionless piston of mass $M$ and cross-sectional area $A$ confines the gas. The system is placed in a region where the downward gravitational acceleration varies with height $y$ from the base of the cylinder as $g(y) = g_0 \frac{y}{h_0}$ . Initially, the piston is in equilibrium at a height $h_0$ . The piston is then given an initial upward velocity $v_0$ . Assuming the expansion of the gas is reversible and adiabatic, what must be the value of $v_0$ so that the maximum height reached by the piston is $8h_0$ ? (Assume the space above the piston is a vacuum).

Options

A.
$\frac{9\sqrt{3}}{2} \sqrt{g_0 h_0}$
B.
$\frac{\sqrt{47}}{2} \sqrt{g_0 h_0}$
C.
$\frac{3\sqrt{14}}{2} \sqrt{g_0 h_0}$
D.
$\frac{3\sqrt{29}}{2} \sqrt{g_0 h_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Conservative Force Field|Adiabatic/Isothermal Processes via KTG|First Law Application

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of the First Law of Thermodynamics with the Work-Energy Theorem in a non-uniform conservative field.

A common lethal mistake is treating the gravitational field as constant, leading to an incorrect potential energy change (Trap Option B).

Furthermore, students often fail to integrate the pressure-volume work as a function of the field-induced boundary motion, or they completely ignore the thermodynamic work done by the gas (Trap Option C).

The problem forces a transition from standard kinetic theory to generalized mechanics by requiring simultaneous integration of thermodynamic work and conservative field work, anchored by a boundary condition dictated by the local gravitational field at equilibrium.

Quick Facts About This Question

Question Information

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