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Question #42446

Difficulty: 8/10 | 🟢 Free

The Question

A uniform thin spherical shell of mass $M$ and radius $R$ is initially at rest in free space. A point mass $m$ is released from rest at an infinite distance and moves towards the shell along a straight line passing through its center. The shell has a tiny hole that allows the point mass to enter its interior. Let $v_{\text{rel}}$ be the relative speed of the point mass with respect to the shell when it is at a distance $R/2$ from the center. Let $\Phi$ be the magnitude of the net gravitational flux through a spherical surface of radius $2R/3$ centered at the shell's center, at this same instant. Which of the following represents the pair $(v_{\text{rel}}, \Phi)$ ?

Options

A.
$\left( \sqrt{\frac{2G(M+m)}{R}}, 4\pi G m \right)$
B.
$\left( \sqrt{\frac{4G(M+m)}{R}}, 4\pi G m \right)$
C.
$\left( \sqrt{\frac{2GM}{R}}, 4\pi G (M+m) \right)$
D.
$\left( \sqrt{\frac{4GM}{R}}, 0 \right)$

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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

Newton’s Third Law Pairing|Shell Theorem (Internal vs External)|Gravitational Self-Energy

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the Shell Theorem, two-body energy conservation, and Gauss's Law for gravitation.

A common trap is assuming the gravitational potential continues to decrease as $-GMm/r$ inside the shell, leading to an incorrect relative velocity (Option B).

Students also frequently forget to use the reduced mass for a two-body system where both objects are free to move, incorrectly using just $M$ (Options C and D).

Finally, the flux calculation requires recognizing that only the enclosed point mass contributes to the net flux, while the outer shell contributes nothing, reinforcing the analogy between gravitational and electric flux.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Newton’s Third Law Pairing
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free