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Question #43306

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down a rough inclined plane of angle $\alpha$ . Its center is connected to a fixed support at the top of the incline by a uniform heavy spring of mass $m$ and stiffness $k$ , which is parallel to the incline. The cylinder is also coupled to a stationary superconducting coil of inductance $L$ via an ideal, massless electromechanical transducer. The transducer ensures that the magnetic flux $\Phi$ linked to the inductor is directly proportional to the displacement $x$ of the cylinder's center from the spring's unstretched position, given by $\Phi = \beta x$ (where $\beta$ is a positive constant). The circuit has zero electrical resistance and the initial current is zero when $x=0$ . If the cylinder is released from rest at $x=0$ , what is the angular frequency $\omega$ of its oscillations?

Options

A.
$\sqrt{\frac{6(k L + \beta^2)}{L(9 M + 2 m)}}$
B.
$\sqrt{\frac{2(k L + \beta^2)}{L(3 M + 2 m)}}$
C.
$\sqrt{\frac{3(k L + \beta^2)}{L(3 M + m)}}$
D.
$\sqrt{\frac{2(k L + \beta^2)}{3 L M}}$

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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

Energy Stored in Inductor|Spring Potential Energy|Pure Rolling Condition

Why This Question Is Tricky

Common Mistakes

This question tests multiple advanced concepts simultaneously, primarily focusing on the correct formulation of effective mass and effective stiffness in a conservative system.

Students often fall into three specific traps: (1) misapplying the kinetic energy of a continuous system by either ignoring the heavy spring or incorrectly using its full mass $m$ instead of the integrated effective mass $m/3$ ; (2) forgetting the rotational kinetic energy component of the rolling cylinder, which adds $M/2$ to the effective mass; and (3) failing to recognize that the electromechanical coupling (back-EMF) acts as a conservative non-mechanical restoring force, storing magnetic potential energy $U_L = \beta^2 x^2 / (2L)$ that effectively increases the system's stiffness.

By requiring the synthesis of rigid body dynamics, continuous mass systems, and electromagnetic induction, the problem rigorously evaluates the student's ability to construct a comprehensive energy conservation equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Energy Stored in Inductor
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free