Before You Begin

Question #42649

Difficulty: 8/10 | 🟢 Free

The Question

A spherical cloud of dust of radius $R$ (in $\text{m}$ ) and total mass $M$ (in $\text{kg}$ ) is held together by its own gravity. The mass density of the cloud varies radially as $\rho(r) = c r$ , where $c$ is a positive constant in $\text{kg}\cdot\text{m}^{-4}$ and $r$ is the distance from the center in $\text{m}$ . Let $V_c$ be the gravitational potential at the center of the cloud in $\text{J}\cdot\text{kg}^{-1}$ (taking the potential at infinity to be zero), and $U$ be the total gravitational self-energy of the cloud in $\text{J}$ . Which of the following expressions correctly relates $U$ , $M$ , and $V_c$ ?

Options

A.
$U = \frac{3}{7} M V_c$
B.
$U = \frac{6}{7} M V_c$
C.
$U = \frac{2}{5} M V_c$
D.
$U = \frac{1}{2} M V_c$

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

Non-conducting Charged Sphere|Gravitational Potential Energy (System)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the derivation of gravitational potential and self-energy for a non-uniform mass distribution, specifically targeting sign errors in potential gradients.

A common mistake is incorrectly calculating the central potential by adding the integral of the field to the surface potential ( $V_c = V(R) + \int g dr = -\frac{2GM}{3R}$ ), ignoring that gravitational potential must become more negative towards the center.

This sign error leads to the trap answer $U = \frac{6}{7} M V_c$ .

Another trap is assuming the uniform density relation $U = \frac{2}{5} M V_c$ .

The correct approach requires $V_c = V(R) - \int g dr = -\frac{4GM}{3R}$ , yielding $U = \frac{3}{7} M V_c$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Non-conducting Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free