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Question #42484

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid sphere of initial mass $M$ and radius $R$ slowly accretes dust from infinity. During the accretion process, the sphere maintains a constant uniform density $\rho$ at all times. The accretion continues until the radius of the sphere becomes $2R$ . What is the total work done by the gravitational forces of the system during this specific accretion phase?

Options

A.
$\frac{93GM^2}{5R}$
B.
$\frac{96GM^2}{5R}$
C.
$\frac{9GM^2}{2R}$
D.
$\frac{147GM^2}{10R}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Potential Energy Reference|Gravitational Self-Energy|Gravitational Potential Energy (System)

Why This Question Is Tricky

Common Mistakes

Students typically treat the self-energy of a system as a static scalar or incorrectly assume the core mass remains constant during accretion.

A common fallacy is calculating the work done by integrating $dW = G M_{initial} dm / r$ , ignoring that the gravitating mass $m$ itself increases as $r^3$ due to the constant density constraint.

Another trap is calculating the self-energy of the final $8M$ sphere but failing to subtract the initial self-energy of the $M$ sphere, yielding $96GM^2/5R$ .

This question forces students to recognize that systemic self-energy is a dynamic work-done quantity dependent on the continuous accumulation of mass, requiring the integration of $dW = G m(r) dm / r$ over the specific accretion interval.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free