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Question #42286

Difficulty: 8/10 | 🟢 Free

The Question

A satellite of mass $M_s$ is in a circular orbit of radius $R_0$ around a planet of mass $M$ . The satellite is tidally locked, maintaining a constant orientation relative to the planet. Inside the satellite, a smooth straight tube is aligned along the radial direction. Two pistons, each of mass $m$ ( $m \ll M_s$ ), are placed in the tube symmetrically on opposite sides of the satellite's center of mass. Between the pistons is $n$ moles of an ideal gas maintained at a constant temperature $T$ . An internal mechanical system exerts forces on the pistons to ensure they move quasi-statically from a distance $L$ to $2L$ from the center of mass. What is the total work done by this mechanical system on the pistons? (Let $R$ be the ideal gas constant)

Options

A.
$- \frac{9 G M m L^2}{R_0^3} - n R T \ln 2$
B.
$- \frac{3 G M m L^2}{R_0^3} - n R T \ln 2$
C.
$- \frac{12 G M m L^2}{R_0^3} - n R T \ln 2$
D.
$- n R T \ln 2$

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

Weightlessness in Orbit|Isothermal Process Work|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in a rotating, non-inertial reference frame, specifically highlighting the trap of using the inertial frame's energy without accounting for the work done by constraint forces.

Students often incorrectly assume that the work done by the internal mechanism simply equals the change in the system's total inertial mechanical energy ( $\Delta E = -3GMmL^2/R_0^3$ ) minus the gas work.

However, because the satellite is tidally locked, the guide tube must exert a tangential force (canceling the Coriolis force) that does non-conservative work in the inertial frame to increase the system's angular momentum.

Analyzing the system in the rotating frame using the effective tidal-centrifugal potential elegantly avoids this trap and reveals the true radial work required.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Weightlessness in Orbit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free