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Question #80943

Difficulty: 8/10 | 🟢 Free

The Question

Assertion (A): A composite solid cylinder of radius $R$ and total mass $M$ is formed by taking a uniform cylinder of density $\rho$ and replacing a cylindrical portion of radius $R/2$ (centered at a distance $R/2$ from the geometric axis) with a material of density $3\rho$ . When placed on a rough horizontal surface and slightly displaced from its stable equilibrium, it rolls without slipping with a time period $T = 3\pi \sqrt{\frac{3R}{g}}$ , where $g$ is the acceleration due to gravity.

Reason (R): The restoring torque about the instantaneous axis of rotation is provided by the weight of the cylinder. The moment of inertia about the contact point can be calculated using the parallel axis theorem as $I_P = I_C + MR^2$ , where $I_C$ is the moment of inertia about the geometric center.

Options

A.
Both (A) and (R) are correct and (R) is the correct explanation of (A)
B.
Both (A) and (R) are correct but (R) is not the correct explanation of (A)
C.
(A) is correct but (R) is incorrect
D.
(A) is incorrect but (R) is correct

Check Your Answer

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

Simple Pendulum Small Angle Approximation|Torque about non-fixed axes|Calculus (Integration)

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Common Mistakes

This question tests the rigorous application of the parallel axis theorem and the dynamics of rolling bodies with non-uniform mass distributions.

A common trap students fall into is applying the parallel axis theorem ( $I = I_0 + Md^2$ ) between two arbitrary parallel axes.

The theorem strictly requires that one of the axes must pass through the center of mass.

By shifting the center of mass away from the geometric center, the problem forces students to first find $I_{cm}$ before finding the moment of inertia about the instantaneous axis of rotation (the contact point).

Reason (R) explicitly presents this flawed application, making it a false statement, while Assertion (A) provides the correct time period derived from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Simple Pendulum Small Angle Approximation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free