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Question #43500

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylinder of mass $M$ and radius $R$ is mounted on a fixed, frictionless vertical axis. A uniform heavy rope of mass $m$ is wrapped around the cylinder, subtending a total angle $\theta_0 = \pi$ at the center. The coefficient of static friction between the rope and the cylinder is $\mu_s = \frac{\ln 2}{\pi}$ . A constant horizontal force $F$ is applied tangentially to one end of the rope, while the other end is free. What is the maximum mass $M$ of the cylinder such that the rope does not slip at the instant the force is applied (when the angular velocity is zero)?

Options

A.
$2m \left( \frac{1 - \ln 2}{\ln 2} \right)$
B.
$2m \left( \frac{\ln 2}{1 - \ln 2} \right)$
C.
$m \left( \frac{1 - \ln 2}{\ln 2} \right)$
D.
$2m$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Tension in Massless vs Massive Strings|Pure Rolling Condition|Kinetic Friction Transition

Why This Question Is Tricky

Common Mistakes

This question tests the transition from lumped-mass approximations to rigorous differential analysis in rotational dynamics.

The 'Lethal Collision' occurs when students misapply the standard capstan equation ( $T_2 = T_1 e^{\mu \theta}$ ), which assumes a massless string.

Because the free end has zero tension, the massless formula incorrectly predicts zero tension everywhere.

Alternatively, students may assume uniform tension to calculate total friction, ignoring the exponential coupling between normal force and tension.

By treating the cable as a continuous mass distribution, the net torque equation requires setting up a differential inequality ( $dT/d\theta - \mu_s T \le \lambda R^2 \alpha$ ) that accounts for the tension gradient created by the string's own mass accelerating.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Tension in Massless vs Massive Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free