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Question #42278

Difficulty: 8/10 | 🟢 Free

The Question

A hypothetical spherical planet of radius $R$ is formed by assembling dust particles from infinity. The 'bare' density profile (defined as the sum of the rest masses of the constituent dust particles per unit volume) is given by $\rho(r) = \frac{k}{r}$ for $0 < r \le R$ , where $k$ is a positive constant. Due to mass-energy equivalence, the gravitational binding energy released during formation reduces the planet's effective gravitational mass. Assuming the binding energy is calculated using the bare mass distribution, what is the escape velocity for a small test mass launched from the surface of this planet? ( $G$ is the universal gravitational constant, $c$ is the speed of light in vacuum)

Options

A.
$\sqrt{4\pi k G R \left( 1 - \frac{4 \pi k G R}{3 c^2} \right)}$
B.
$\sqrt{4\pi k G R \left( 1 + \frac{4 \pi k G R}{3 c^2} \right)}$
C.
$\sqrt{4\pi k G R \left( 1 - \frac{2 \pi k G R}{3 c^2} \right)}$
D.
$\sqrt{4\pi k G R}$

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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

Escape Velocity Independence|Mass Defect vs. Binding Energy|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of classical gravitation and relativistic mass-energy equivalence.

A common trap is calculating the escape velocity using only the integrated bare mass (Option D), ignoring that the negative gravitational binding energy contributes to the total mass-energy of the system.

By forcing students to calculate the binding energy of a variable-density sphere and subtract its mass equivalent ( $\Delta M = |U|/c^2$ ) from the bare mass, it evaluates their understanding that effective inertial/gravitational mass in high-gravity systems is reduced by the energy radiated away during formation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Escape Velocity Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free