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Question #42400

Difficulty: 8/10 | 🟢 Free

The Question

A non-uniform rod of mass $M$ and length $L$ is pivoted freely at its upper end $O$ . The linear mass density of the rod varies as $\lambda(x) = cx$ , where $x$ is the distance from the pivot $O$ and $c$ is a positive constant. The rod is initially hanging vertically at rest. A horizontal impulsive force $J$ is applied at the bottom end of the rod. What is the minimum value of $J$ required so that the rod completes a full vertical circle about the pivot $O$ ?

Options

A.
$2M \sqrt{\frac{gL}{3}}$
B.
$M \sqrt{\frac{10gL}{3}}$
C.
$M \sqrt{\frac{8gL}{3}}$
D.
$M \sqrt{5gL}$

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

Minimum Speed for Looping|Impulsive Torque and Angular Momentum|System of Particles and Rotational Motion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of angular impulse, rotational inertia, and energy conservation for a non-uniform rigid body.

The primary trap is incorrectly treating the rod as a point mass at its center of mass and applying standard particle formulas like $v = \sqrt{5gR}$ or $v = \sqrt{4gR}$ to find the impulse via $J = Mv$ .

Students must recognize that the mass distribution requires integration to find both $x_{cm}$ and $I_O$ , and that the minimum condition for a rigid pivoted rod to complete a vertical circle is reaching the top with zero angular velocity, unlike a string which requires a minimum tension.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Minimum Speed for Looping
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free