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Question #42432

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylinder of mass $M$ and radius $R$ is free to rotate about its fixed horizontal axis. Initially, it rotates with angular velocity $\omega_0$ . A small block of mass $m$ is placed at the top of the cylinder ( $\theta = 0$ ). An external mechanism forces the block to slide down the surface of the cylinder, in the direction of its rotation, at a constant speed $v_0 = \sqrt{gR/2}$ relative to the ground. The mechanism only exerts a tangential force on the block. The coefficient of kinetic friction between the block and the cylinder is $\mu$ . Assume the cylinder's surface always moves faster than the block ( $\omega R > v_0$ ). Find the angular velocity of the cylinder when the block reaches an angular position $\theta = \pi/6$ from the top.

Options

A.
$\omega_0 - \frac{\mu m}{M} \sqrt{\frac{2g}{R}} \left(\frac{6-\pi}{6}\right)$
B.
$\omega_0 - \frac{\mu m}{M} \sqrt{\frac{2g}{R}}$
C.
$\omega_0 - \frac{\mu m}{M} \sqrt{\frac{2g}{R}} \left(\frac{\pi}{3}\right)$
D.
$\omega_0 - \frac{\mu m}{M} \sqrt{\frac{2g}{R}} \left(\frac{6-\pi}{12}\right)$

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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

Normal Force at Angles|Angular Momentum Conservation|Parallel and Perpendicular Axes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate a non-conservative frictional torque over a path where the normal force varies continuously.

The primary trap is assuming a constant normal force (e.g., $N = mg$ ) or ignoring the centripetal acceleration requirement ( $m v^2/R$ ) in the radial force balance, which leads to incorrect torque integration.

Furthermore, students must correctly relate the spatial variable $\theta$ to time $t$ using the kinematic constraint of constant speed to evaluate the angular impulse, bridging point-mass dynamics with rigid-body rotational mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Normal Force at Angles
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free