Before You Begin

Question #42388

Difficulty: 8/10 | 🟢 Free

The Question

A spherical dust cloud of mass $M$ and radius $R$ initially has a uniform density and rotates with an angular velocity $\omega_0$ . It undergoes gravitational collapse, radiating away energy, until it settles into a new spherical shape of radius $R/2$ . In this final state, the cloud has a non-uniform density profile given by $\rho(r) = \frac{c}{r}$ (where $c$ is a constant and $r$ is the distance from the center). Assuming no external torques act on the system, what is the net energy radiated away during this process?

Options

A.
$\frac{11GM^2}{15R} - \frac{19MR^2\omega_0^2}{25}$
B.
$\frac{11GM^2}{15R} - \frac{3MR^2\omega_0^2}{5}$
C.
$\frac{3GM^2}{5R} - \frac{19MR^2\omega_0^2}{25}$
D.
$\frac{3GM^2}{5R} - \frac{3MR^2\omega_0^2}{5}$

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

Work-Energy Theorem for Rotation|Gravitational Self-Energy|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between the Work-Energy Theorem, Gravitational Self-Energy, and Angular Momentum conservation during mass redistribution.

The primary trap is assuming the moment of inertia scales simply as $MR^2$ without accounting for the non-uniform density profile $\rho(r) \propto 1/r$ , which alters both the final moment of inertia and the gravitational self-energy.

Students often incorrectly apply the uniform sphere formulas $I = \frac{2}{5}MR^2$ and $U = -\frac{3GM^2}{5R}$ to the final state, failing to integrate over the new mass distribution.

This forces the realization that rotational kinetic energy gain is explicitly balanced by the change in internal gravitational potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Work-Energy Theorem for Rotation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free