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Question #42896

Difficulty: 8/10 | 🟢 Free

The Question

A spherical cloud of dust of total mass $M$ and radius $R$ has a non-uniform mass density given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $0 \le r \le R$ , where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. What is the magnitude of the gravitational self-energy of this dust cloud? (Here $G$ is the universal gravitational constant)

Options

A.
$\frac{26}{35} \frac{G M^2}{R}$
B.
$\frac{3}{5} \frac{G M^2}{R}$
C.
$\frac{2}{5} \frac{G M^2}{R}$
D.
$\frac{1}{2} \frac{G M^2}{R}$

Check Your Answer

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

Superposition Principle|Gravitational Self-Energy|Variable Mass Systems

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Superposition Principle and Gauss's Law for gravitation in non-uniform mass distributions.

A common trap is assuming the Shell Theorem allows treating the entire mass $M$ as a point mass for internal energy calculations, or incorrectly assuming a uniform density result ( $3GM^2/5R$ ).

Students must recognize that self-energy requires integrating the potential of the already-assembled inner sphere $M(r)$ interacting with the incoming differential mass shell $dm$ .

Failing to correctly express both $M(r)$ and $dm$ as functions of $r$ before integrating leads to dimensionally similar but numerically incorrect prefactors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Superposition Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free