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Question #42331

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal tube of moment of inertia $I_0$ rotates freely about a vertical axis passing through its center. A particle of mass $m$ can slide without friction inside the tube. The particle is connected to the axis of rotation by a light spring of spring constant $k$ and natural length zero. The total angular momentum of the system is $L$ and is strictly conserved. The particle is in a stable dynamic equilibrium at a radial distance $r_0 > 0$ . What is the angular frequency $\Omega$ of small radial oscillations of the particle about $r_0$ ?

Options

A.
$2 r_0 \sqrt{ \frac{k}{I_0 + m r_0^2} }$
B.
$\sqrt{ \frac{k}{m} }$
C.
$\sqrt{ \frac{2k}{m} }$
D.
$r_0 \sqrt{ \frac{k}{I_0 + m r_0^2} }$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Equilibrium Stability|Work-Energy Theorem for Rotation|Gyroscopic Stability|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between static equilibrium under a driven constant angular velocity $\omega$ and dynamic equilibrium under a conserved angular momentum $L$ .

Students often fall into the trap of treating $\omega$ as a constant parameter during oscillations, which incorrectly yields a neutral stability condition ( $\Omega = 0$ ) because the centrifugal force perfectly balances the spring force at all radii.

By forcing the use of the effective potential $V_{eff}(r) = V(r) + L^2/2I(r)$ , the problem evaluates the understanding of gyroscopic stiffening.

Conservation of $L$ dictates that as the particle moves outward, the rotational inertia increases, causing the angular velocity and the outward centrifugal force to drop.

This coupling provides the actual restoring force for the oscillations, a core concept in Lagrangian mechanics and rigid-body dynamics.

Quick Facts About This Question

Question Information

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