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Question #43673

Difficulty: 8/10 | 🟢 Free

The Question

A hypothetical spherical celestial body is composed of $A$ identical particles, each of mass $m_0$ . The density of the body varies with the radial distance $r$ from its center as $\rho(r) = k r$ , where $k$ is a universal constant for this type of matter. A reference body made of this exact same matter containing $A_0$ particles has a known radius $R_0$ . What is the magnitude of the gravitational self-energy of the body containing $A$ particles? (Assume $G$ is the universal gravitational constant).

Options

A.
$\frac{4 G m_0^2 A_0^{1/4}}{7 R_0} A^{7/4}$
B.
$\frac{3 G m_0^2 A_0^{1/4}}{5 R_0} A^{7/4}$
C.
$\frac{4 G m_0^2 A_0^{1/3}}{7 R_0} A^{5/3}$
D.
$\frac{3 G m_0^2 A_0^{1/3}}{5 R_0} A^{5/3}$

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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

Nuclear Radius Scaling|Gravitational Self-Energy|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive scaling laws and self-energy for non-uniform mass distributions, bypassing rote memorization.

The primary trap is twofold: first, students may incorrectly assume the standard nuclear radius scaling $R \propto A^{1/3}$ (which only applies to uniform density matter) instead of deriving $R \propto A^{1/4}$ from the given $\rho(r) \propto r$ profile.

Second, they may blindly apply the uniform sphere self-energy formula $U = -3GM^2/5R$ instead of integrating $dU = -Gm(r)dm/r$ for the specific density profile.

Options C and D bait the $A^{1/3}$ scaling, while Options B and D bait the $3/5$ uniform sphere coefficient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free