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Question #43515

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid spherical satellite of mass $m$ and radius $a$ ( $a \ll r_0$ ) is initially in a circular geostationary orbit of radius $r_0$ around the Earth (mass $M$ ). Initially, the satellite is irrotational relative to the distant stars. An internal motor, powered by onboard batteries, spins the satellite up until it achieves a 'pure rolling' state along its orbit. In this state, its spin angular velocity $\omega$ and its new orbital speed $v$ satisfy $v = \omega a$ , with the spin and orbital angular momentum vectors being parallel. Assuming no mass is ejected and no external torques act on the satellite about the Earth's center, find the approximate change in the orbital radius $\Delta r = r - r_0$ .

Options

A.
$-\frac{4}{5}a$
B.
-2a
C.
$-\frac{2}{5}r_0$
D.
$+\frac{4}{5}a$

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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

Orbital Velocity Energy Balance|Geostationary Satellite Constraints|Pure Rolling Condition

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the conservation of mechanical energy and the conservation of angular momentum in orbital mechanics.

A common trap is assuming that the total mechanical energy of the satellite is conserved, leading students to incorrectly balance the newly acquired rotational kinetic energy with a massive drop in orbital potential energy (resulting in Option C).

The correct physical framework requires recognizing that the internal motor converts chemical energy into mechanical energy, breaking energy conservation, while the absence of external torques guarantees the conservation of angular momentum.

The coupling of the internal spin to the orbital path forces a slight orbital decay to compensate for the acquired spin angular momentum.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Orbital Velocity Energy Balance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free