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Question #43448

Difficulty: 8/10 | 🟢 Free

The Question

A small spinning top of mass $M$ is pivoted at the origin $(0,0,0)$ of a Cartesian coordinate system. The top has a moment of inertia $I_0$ about its symmetry axis and spins with a very high angular velocity $\omega$ . Its center of mass is located at a distance $l$ from the pivot along the symmetry axis. The top possesses a magnetic dipole moment of magnitude $m$ directed along its symmetry axis. Acceleration due to gravity is $g$ in the $-z$ direction. Two infinitely long straight wires are placed parallel to the $z$ -axis. Wire 1 passes through $(0, d, 0)$ and carries a current $I$ in the $+z$ direction. Wire 2 passes through $(d, 0, 0)$ and carries a current $I$ in the $-z$ direction. Assume the dimensions of the top are much smaller than $d$ . If the top undergoes steady precession, what is the magnitude of its precession angular velocity $\Omega_p$ ?

Options

A.
$\frac{1}{I_0 \omega} \sqrt{M^2 g^2 l^2 + \frac{m^2 \mu_0^2 I^2}{2 \pi^2 d^2}}$
B.
$\frac{1}{I_0 \omega} \left( M g l + \frac{m \mu_0 I}{\sqrt{2} \pi d} \right)$
C.
$\frac{1}{I_0 \omega} \sqrt{M^2 g^2 l^2 + \frac{m^2 \mu_0^2 I^2}{\pi^2 d^2}}$
D.
$\frac{1}{I_0 \omega} \sqrt{M^2 g^2 l^2 + \frac{m^2 \mu_0^2 I^2}{4 \pi^2 d^2}}$

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Correct Answer

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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

Superposition of Magnetic Fields|Gyroscopic Stability|Torque about non-fixed axes|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of magnetostatics and rigid body dynamics, specifically targeting the misconception that gyroscopic precession always occurs around the vertical axis.

Students often calculate the precession rates due to gravity and the magnetic field separately and add them algebraically (scalar addition).

The trap lies in the non-fixed axis: because the magnetic field is horizontal, the net torque vector is not confined to the horizontal plane.

Consequently, the axis of steady precession tilts, requiring a full vector formulation of Euler's equation ( $\frac{d\vec{L}}{dt} = \vec{\tau}$ ) to find the true precession angular velocity vector $\vec{\Omega}_p$ .

Furthermore, it tests the correct vector superposition of magnetic fields from multiple wires.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Superposition of Magnetic Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free