Before You Begin

Question #42260

Difficulty: 8/10 | 🟢 Free

The Question

An inner spherical shell of radius $R$ has a surface mass density $\sigma_1(\theta) = \sigma_0 (1 + 3\cos^2 \theta)$ . An outer concentric spherical shell of radius $2R$ has a surface mass density $\sigma_2(\theta) = \sigma_0 (1 - \cos^2 \theta)$ , where $\theta$ is the polar angle. A particle of mass $m$ is placed at a distance $r = \frac{3}{2}R$ from the center on the polar axis ( $\theta = 0$ ). Find the magnitude of the net gravitational force on the particle.

Options

A.
$\frac{716}{135} \pi G \sigma_0 m$
B.
$\frac{608}{135} \pi G \sigma_0 m$
C.
$\frac{32}{9} \pi G \sigma_0 m$
D.
$\frac{224}{27} \pi G \sigma_0 m$

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

Gravitational Field Variation (Depth vs Height)|Symmetry and Virtual Work|Potential of Concentric Shells

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the Shell Theorem and the misapplication of the 'internal mass' rule.

Students commonly assume that an external spherical shell exerts zero gravitational force on a mass inside it, which is only true for uniform shells.

By requiring the Principle of Virtual Work ( $F = -\nabla \Phi$ ) to find the force, the problem reveals that the outer shell's quadrupole moment ( $P_2$ term) creates a spatially varying potential in its interior.

Ignoring the dynamic linkage between the external shell's potential and the particle's radial displacement leads to omitting the outer shell's force contribution, trapping students into selecting the partial force answer.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Field Variation (Depth vs Height)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free