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Question #42251

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of mass $M$ and radius $r$ rolls without slipping on the inside surface of a fixed hollow cylinder of radius $R$ ( $R > r$ ). The motion is planar, and $\theta$ represents the small angular displacement of the sphere's center from the vertical. A constant rolling resistance torque $\tau_0$ opposes the rotation of the sphere about its own center. If the sphere is released from rest at an initial angle $\theta_0$ (where $\theta_0 > \frac{\tau_0}{Mgr}$ ), what is the square of the maximum angular velocity ( $\dot{\theta}_{max}^2$ ) of the sphere's center during its first downward swing?

Options

A.
$\frac{5g}{7(R-r)} \left( \theta_0 - \frac{\tau_0}{Mgr} \right)^2$
B.
$\frac{g}{R-r} \left( \theta_0 - \frac{\tau_0}{Mgr} \right)^2$
C.
$\frac{5g}{7(R-r)} \left( \theta_0 - \frac{\tau_0}{Mg(R-r)} \right)^2$
D.
$\frac{g}{R-r} \left( \theta_0 - \frac{\tau_0}{Mg(R-r)} \right)^2$

Check Your Answer

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

Compound Pendulum Period|Work Done by Friction|Pure Rolling Condition

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Work-Energy theorem and Newtonian mechanics to a compound pendulum system experiencing non-conservative rolling resistance.

It exploits two major traps.

First, students often treat the sphere as a point mass, neglecting its moment of inertia ( $I = 2/5 Mr^2$ ), which incorrectly alters the effective mass term and leads to an incorrect natural frequency (Option B).

Second, students incorrectly calculate the work done by the rolling resistance torque by assuming the angle of rotation is simply $\theta$ instead of the true rotation angle $\phi = \frac{R-r}{r}\theta$ dictated by the pure rolling constraint, leading to an incorrect equilibrium shift (Options C and D).

Successfully navigating this requires setting up the correct differential equation for energy dissipation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Compound Pendulum Period
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free