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Question #43676

Difficulty: 8/10 | 🟢 Free

The Question

While standard nuclear models often assume density invariance (uniform density $\rho \approx \text{constant}$ ), a more refined classical model of a heavy nucleus accounts for a diffuse surface. Consider a spherical nucleus of total mass $M$ and radius $R$ whose mass density varies radially as $\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right)$ for $r \le R$ , where $\rho_0$ is the central density. What is the magnitude of the gravitational self-energy of this nucleus? (Assume $G$ is the universal gravitational constant).

Options

A.
$\frac{5GM^2}{7R}$
B.
$\frac{3GM^2}{5R}$
C.
$\frac{4GM^2}{7R}$
D.
$\frac{2GM^2}{3R}$

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

Nuclear Density Invariance|Gravitational Field Variation (Depth vs Height)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from memorized standard results to first-principles integration for continuous mass distributions.

The cognitive trap lies in the 'Nuclear Density Invariance' anchor; students often reflexively assume a uniform density for nuclei and apply the standard formula for the gravitational self-energy of a uniform sphere ( $U = 3GM^2/5R$ ).

By introducing a radially varying density profile $\rho(r) = \rho_0(1 - r^2/R^2)$ , the problem invalidates this assumption.

To succeed, students must correctly apply the principle of gravitational potential energy by calculating the enclosed mass $m(r)$ and integrating $dU = -G m(r) dm / r$ , avoiding algebraic shortcuts.

Quick Facts About This Question

Question Information

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