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Question #42262

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of mass $M_0$ and radius $R$ is fixed in space. A thick spherical shell of inner radius $R$ and outer radius $2R$ , having a total mass $M_s$ , is slowly assembled around the sphere by bringing particles from infinity. The density of the assembled shell varies with radial distance $r$ from the center as $\rho(r) = \frac{c}{r}$ , where $c$ is a positive constant. What is the work done by the external agent in assembling the shell?

Options

A.
$-\frac{G M_s}{27 R} (18 M_0 + 8 M_s)$
B.
$-\frac{G M_s}{27 R} (18 M_0 + 4 M_s)$
C.
$-\frac{G M_s}{14 R} (9 M_0 + 8 M_s)$
D.
$-\frac{G M_s}{27 R} (9 M_0 + 8 M_s)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Gravitational Potential Reference|Gravitational Self-Energy|Integration

Why This Question Is Tricky

Common Mistakes

This problem targets the common misconception where students memorize the self-energy formulas for uniform spherical shells or solid spheres and attempt to apply them directly, ignoring the radial dependence of the density.

Furthermore, it tests the ability to correctly set up the differential work equation $dW = V(r)dm$ by accounting for the dynamic shift in the reference potential caused by both the partially assembled shell and the pre-existing central mass $M_0$ .

It forces the derivation of self-energy from first principles, effectively separating rote memorization from a deep conceptual understanding of the work-energy theorem applied to continuous, non-uniform mass distributions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Potential Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free