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Question #42326

Difficulty: 8/10 | 🟢 Free

The Question

A thin cylindrical rod of length $L$ and uniform cross-sectional area $A$ has a non-uniform density given by $\rho(x) = \rho_0 \left(1 + \frac{x}{L}\right)$ , where $x$ is the distance from one end ( $x=0$ ). The rod is completely submerged in a large tank of ideal liquid of uniform density $\rho_0$ . The rod is initially held at rest in a horizontal position at a sufficient depth and then released. Assuming viscous drag is negligible, find the angular velocity $\omega$ of the rod when it reaches the vertical position for the first time.

Options

A.
$\sqrt{\frac{12g}{13L}}$
B.
$\sqrt{\frac{8g}{9L}}$
C.
$\sqrt{\frac{4g}{3L}}$
D.
$\sqrt{\frac{6g}{13L}}$

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

Mechanical Energy Conservation|Archimedes' Buoyancy Dynamics|Integration

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume the rod rotates about its geometric center or one of its ends, failing to decouple the translational and rotational dynamics.

Furthermore, they frequently misidentify the work done by the buoyant force, treating it as a non-conservative force or failing to recognize that in the CM frame, the pseudo-forces do zero work, leaving buoyancy as the sole contributor to rotational kinetic energy.

This question rigorously tests the application of the work-energy theorem in an accelerating non-inertial frame and the distinction between the center of mass and center of buoyancy in non-uniform bodies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Mechanical Energy Conservation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free