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Question #42295

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The Question

A small block of mass $m$ is attached to one end of a uniform wire of natural length $L_0$ , cross-sectional area $A$ , and Young's modulus $Y$ . The other end of the wire is fixed to a vertical axis. The block rests on a smooth horizontal surface and is constrained to move along a radial groove in a horizontal disc. The disc is rotated by a motor at a constant angular velocity $\omega$ . Initially, the block is held at a distance $L_0$ from the axis and then released. Due to internal friction in the wire, the block eventually comes to rest relative to the disc. Assuming $YA > m \omega^2 L_0$ , what is the total energy dissipated as heat due to internal friction?

Options

A.
$\frac{m^2 \omega^4 L_0^3}{2(YA - m \omega^2 L_0)}$
B.
$\frac{m^2 \omega^4 L_0^3}{YA - m \omega^2 L_0}$
C.
$\frac{m^2 \omega^4 L_0^3}{2 YA}$
D.
$\frac{m^2 \omega^4 L_0^3}{2(YA + m \omega^2 L_0)}$

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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

Young's Modulus Sensitivity|Spring Potential Energy|Non-Uniform Circular Motion

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in a non-inertial rotating frame, specifically addressing the trap of ignoring the work done by the motor or miscalculating the centrifugal potential energy.

Students often incorrectly assume the energy dissipated is simply the elastic potential energy (1/2 k x^2) or fail to account for the fact that the centrifugal force increases with radial distance, leading to a coupled equilibrium state.

By requiring the calculation of internal friction dissipation, it forces the use of the effective potential U_eff = U_elastic + U_centrifugal, highlighting that the motor must do extra work in the ground frame to maintain constant angular velocity as the moment of inertia increases.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Young's Modulus Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free