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Question #42338

Difficulty: 8/10 | 🟢 Free

The Question

A non-uniform rod of mass $M$ and length $L$ lies at rest on a smooth horizontal surface. Its linear mass density varies as $\lambda = kx$ , where $x$ is the distance from end A. A small particle of mass $M$ moving with velocity $v_0$ perpendicular to the rod strikes it at end B ( $x=L$ ). If the coefficient of restitution for the collision is $e = \frac{1}{3}$ , find the velocity of the particle immediately after the collision.

Options

A.
$\frac{2v_0}{3}$
B.
$\frac{v_0}{3}$
C.
$\frac{11v_0}{15}$
D.
$\frac{v_0}{2}$

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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

Coefficient of Restitution|System of Particles and Rotational Motion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of conservation laws and collision mechanics on a non-uniform rigid body.

A common trap is assuming the center of mass is at $L/2$ and $I_{cm} = ML^2/12$ , which ignores the density gradient $\lambda = kx$ .

Furthermore, students often incorrectly apply the coefficient of restitution to the center of mass rather than the specific point of impact, failing to account for the rotational velocity component $\omega d$ at the contact point.

It requires synthesizing calculus-based mass distribution with coupled linear and angular impulse equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Coefficient of Restitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free