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Question #42455

Difficulty: 8/10 | 🟢 Free

The Question

A thin spherical shell of radius $R$ is fixed in deep space. Its surface mass density varies as $\sigma(\theta) = \sigma_0 (1 - \cos \theta)$ , where $\theta \in [0, \pi]$ is the polar angle measured from the positive Z-axis. A small block of mass $m$ rests on a rough flat surface (coefficient of static friction $\mu$ ) fixed exactly at the center of the shell. The surface lies in the X-Y plane. What is the minimum force $F$ required to just move the block?

Options

A.
$\frac{4 \pi \mu G m \sigma_0}{3 \sqrt{1 + \mu^2}}$
B.
0
C.
$\frac{4}{3} \pi \mu G m \sigma_0$
D.
$\frac{2 \pi \mu G m \sigma_0}{\sqrt{1 + \mu^2}}$

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

Minimum Force to Move a Block|Shell Theorem (Internal vs External)

Why This Question Is Tricky

Common Mistakes

This question tests the boundaries of the Shell Theorem and the mechanics of friction optimization.

The primary trap is the reflexive assumption that the gravitational field inside any spherical shell is zero; students must recognize that the Shell Theorem strictly requires spherical symmetry (uniform density).

By introducing an angularly dependent density, the problem forces a first-principles integration of the gravitational field.

The secondary trap lies in the minimum force calculation: students often assume the force must be applied horizontally ( $F = \mu N$ ), missing that pulling at an optimal angle reduces the normal force, introducing the $\sqrt{1+\mu^2}$ factor in the denominator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Minimum Force to Move a Block
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free