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Question #42359

Difficulty: 8/10 | 🟢 Free

The Question

A uniform rod of mass $M$ (in kg) and length $2L$ (in m) is suspended horizontally at its center by a vertical torsion wire of torsional constant $C$ (in N $\cdot$ m/rad). Two small collars, each of mass $m$ (in kg), slide symmetrically outward along the rod. At a particular instant, the rod has an angular displacement $\theta$ (in rad) from its equilibrium position and an angular velocity $\omega$ (in rad/s) in the same sense. At this instant, the collars are at a distance $x$ (in m) from the center and are moving outward with a speed $v$ (in m/s) relative to the rod. Which of the following expressions gives the algebraic tangential force $F_t$ (in N) exerted by the rod on one of the collars in the direction of the angular velocity?

Options

A.
$m \left( \frac{-C\theta x + \frac{2}{3}ML^2v\omega}{\frac{1}{3}ML^2 + 2mx^2} \right)$
B.
$m \left( \frac{-C\theta x - 4mx^2v\omega}{\frac{1}{3}ML^2 + 2mx^2} \right)$
C.
$m \left( \frac{-C\theta x + \frac{2}{3}ML^2v\omega + 4mx^2v\omega}{\frac{1}{3}ML^2 + 2mx^2} \right)$
D.
$m \left( \frac{-C\theta x}{\frac{1}{3}ML^2 + 2mx^2} \right)$

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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

System vs Particle Displacement|Torsional Oscillations|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the decoupling of instantaneous center-of-mass displacement from angular acceleration in a system with variable rotational inertia.

The primary trap is twofold: first, students often incorrectly apply $\tau = I\alpha$ instead of the fundamental $\tau = dL/dt$ , thereby missing the $\frac{dI}{dt}\omega$ term caused by the radial motion of the masses.

Second, when calculating the tangential force on the collar, students frequently forget the Coriolis acceleration ( $2v\omega$ ) in the rotating frame, considering only the Euler acceleration ( $x\alpha$ ).

By requiring the calculation of the internal force, the problem forces the synthesis of the generalized rotational Newton's second law with complete polar kinematics.

Quick Facts About This Question

Question Information

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