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Question #42391

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid cylinder of mass $M=4m$ and radius $R$ is mounted on a frictionless, fixed vertical axis. A projectile of mass $m$ is launched from the ground with an initial speed $u$ at an angle of $60^{\circ}$ to the horizontal. At the highest point of its trajectory, the projectile strikes the rough vertical surface of the cylinder. The vertical plane containing the projectile's trajectory is at a perpendicular distance $b = R/2$ from the cylinder's axis. The coefficient of restitution for the normal component of the collision is $e = 1/2$ . If the projectile stops slipping on the cylinder during the infinitesimal duration of the collision, what is the minimum coefficient of kinetic friction $\mu$ between the projectile and the cylinder?

Options

A.
$\frac{4\sqrt{3}}{27}$
B.
$\frac{2\sqrt{3}}{9}$
C.
$\frac{8\sqrt{3}}{27}$
D.
$\frac{2\sqrt{3}}{27}$

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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

Impulsive Torque and Angular Momentum|Projectile Path Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of translational and rotational constraints during an impulsive collision.

A common trap is assuming the point of impact on the cylinder remains stationary, leading students to incorrectly equate the final tangential velocity of the projectile to zero ( $v_{it} - J_t/m = 0$ ).

The correct approach requires applying the Collision Point Velocity vector equation, recognizing that the impulsive torque imparts an angular velocity to the cylinder during the infinitesimal collision time, thus changing the boundary condition for slipping ( $v_{ft} = \omega R$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Impulsive Torque and Angular Momentum
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free