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Question #42379

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylinder of mass $m$ and radius $r$ rolls without slipping inside a fixed vertical circular track of radius $R = 5r$ . The track exerts a constant rolling friction torque $\tau_r = \frac{1}{\pi} m g r$ on the cylinder, opposing its rotation. To just complete the vertical loop without losing contact at the highest point (vertex), the cylinder must be given a minimum initial velocity $v_0$ at the lowest point. Assuming rolling without slipping is maintained throughout, find $v_0$ .

Options

A.
$2\sqrt{5gr}$
B.
$\sqrt{\frac{56}{3}gr}$
C.
$2\sqrt{7gr}$
D.
$5\sqrt{gr}$

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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

Parallel and Perpendicular Axes|Vertical Circular Motion|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of rigid body dynamics and non-conservative work in a vertical circular path.

A primary trap is failing to recognize that the center of mass moves in a circle of radius $R-r$ , not $R$ , which affects the centripetal condition at the vertex, the potential energy change, and the total rotation angle $\Delta \theta$ .

Additionally, students often miscalculate the kinetic energy by either treating the cylinder as a sliding particle or forgetting to apply the Parallel Axis Theorem for the Instantaneous Axis of Rotation (IAR) at the contact point.

Finally, the non-conservative work requires integrating the constant torque over the cylinder's proper self-rotation angle, punishing those who blindly use the orbital angle $\pi$ .

Quick Facts About This Question

Question Information

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