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Question #42398

Difficulty: 8/10 | 🟢 Free

The Question

A gyroscope consists of a symmetric rotor of mass $m$ and moment of inertia $I$ about its symmetry axis. It is mounted on a massless rod of length $L$ pivoted at the origin $O$ . A point electric dipole of moment $\vec{p}$ is fixed at the rotor's center of mass, aligned with the symmetry axis pointing away from $O$ . The system is placed in a uniform gravitational field $\vec{g} = -g \hat{k}$ and a non-uniform electric field $\vec{E} = E_0 \hat{k} + \beta (x \hat{i} + y \hat{j} - 2z \hat{k})$ , where $\beta > 0$ . The rotor spins with a high angular velocity $\omega_s$ about its axis. Assuming steady precession at a constant angle $\theta$ with the $z$ -axis, what is the precession angular velocity $\Omega$ ?

Options

A.
$\frac{mgL - pE_0 + 6\beta pL \cos\theta}{I\omega_s}$
B.
$\frac{mgL - pE_0 + 3\beta pL \cos\theta}{I\omega_s}$
C.
$\frac{mgL - pE_0}{I\omega_s}$
D.
$\frac{mgL - pE_0 - 2\beta pL \cos\theta}{I\omega_s}$

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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

Gyroscopic Stability|Electric Field Symmetry|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the complete vector formulation of torque on a dipole in a non-uniform electric field coupled with gyroscopic precession.

The lethal trap is assuming the electric torque is simply $\vec{p} \times \vec{E}$ evaluated at the origin or the center of mass.

In a non-uniform field, the spatial gradient induces a net force $\vec{F} = (\vec{p} \cdot \nabla)\vec{E}$ on the dipole.

Because the dipole is located at a distance $L$ from the pivot, this force generates an additional torque $\vec{r} \times \vec{F}$ that is of the exact same order of magnitude as the local dipole torque.

Students who fail to resolve the full vector sum of torques will select the trap options missing the factor of 6.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Gyroscopic Stability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free