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Question #42281

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of radius $R$ and total mass $M$ has a radially varying mass density $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ , where $r$ is the distance from the center. A narrow, smooth radial tunnel is drilled into the sphere. A small particle of mass $m$ is projected from a depth $r = \frac{R}{2}$ inside the tunnel towards the surface. What is the minimum projection speed $v_0$ required for the particle to escape the gravitational pull of the sphere?

Options

A.
$\frac{1}{2}\sqrt{\frac{13GM}{R}}$
B.
$\sqrt{\frac{11GM}{4R}}$
C.
$\sqrt{\frac{3GM}{R}}$
D.
$\frac{1}{2}\sqrt{\frac{15GM}{R}}$

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

Gravitational Potential Energy (System)|Mechanical Energy Conservation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of gravitational potential inside a non-uniform mass distribution.

A common trap for students is to incorrectly assume that the potential inside the sphere can be calculated by treating only the enclosed mass $m(r)$ as a point mass at the center, yielding $V = -Gm(r)/r$ .

This completely ignores the gravitational potential contribution from the outer spherical shells ( $- \int G dm/r'$ ).

The problem forces the explicit derivation of the potential function $V(r)$ via integration of both inner and outer mass elements before applying mechanical energy conservation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Potential Energy (System)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free