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Question #43265

Difficulty: 8/10 | 🟢 Free

The Question

A turntable rotates with a constant angular velocity $\Omega$ about the vertical $Z$ -axis. A vertical frame is rigidly fixed to the turntable. A uniform thin rod of mass $M$ , length $2L$ , and magnetic dipole moment $\mu$ (directed along its length) is pivoted at its center to this frame, such that it can only rotate within the vertical plane of the frame. A uniform magnetic field $\vec{B}$ exists in the vertically upward direction. Let $I$ be the moment of inertia of the rod about its pivot. If the turntable's angular velocity is set to $\Omega = \sqrt{\frac{2\mu B}{I}}$ , what is the angular frequency of small oscillations of the rod about its stable equilibrium position in the rotating frame?

Options

A.
$\sqrt{\frac{3\mu B}{2I}}$
B.
$\sqrt{\frac{\mu B}{I}}$
C.
$\sqrt{\frac{2\mu B}{I}}$
D.
$\sqrt{\frac{5\mu B}{2I}}$

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

Torque and Potential Energy in B-Field|Conservation of Angular Momentum|Frame of Reference Inertial vs Non-Inertial

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze rotational dynamics in a non-inertial reference frame, specifically focusing on the often-neglected centrifugal torque.

A common trap is to treat the system as a simple magnetic pendulum, applying only the magnetic torque $\tau = \vec{\mu} \times \vec{B}$ and completely ignoring the fictitious forces arising from the frame's rotation (which leads to the incorrect Option B).

Even if students remember the centrifugal force, they might incorrectly assume it produces no torque or miscalculate its dependence on the angle $\theta$ .

By setting the rotation rate $\Omega$ high enough to cause a pitchfork bifurcation (making the vertical position unstable), the problem forces students to find the new equilibrium angle and linearize the highly non-linear effective potential to find the oscillation frequency, bridging electromagnetism and advanced classical mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Torque and Potential Energy in B-Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free