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Question #43683

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting rod of mass $M$ (kg) and length $L$ (m) is pivoted at one end and rotates in a horizontal plane. Its free end slides on a frictionless, perfectly conducting circular ring of radius $L$ . A uniform vertical magnetic field $B$ (T) is applied perpendicular to the plane of rotation. A stationary resistor of resistance $R$ ( $\Omega$ ) connects the pivot and the ring. The rod is composed of a radioactive material with a decay constant $\lambda$ (s $^{-1}$ ). The energy released from the radioactive decay is continuously and entirely converted into mechanical power that drives the rotation of the rod. The initial total available decay energy if all nuclei were to decay is $E_0$ (J). Assuming the mass $M$ of the rod remains practically constant, and the rod starts from rest at $t=0$ , at what time $t_m$ (s) will the angular velocity of the rod reach its maximum value?

Options

A.
$\frac{2RM}{3B^2L^2 - 2RM\lambda} \ln\left( \frac{3B^2L^2}{2RM\lambda} \right)$
B.
$\frac{2RM}{B^2L^2 - 2RM\lambda} \ln\left( \frac{B^2L^2}{2RM\lambda} \right)$
C.
$\frac{RM}{6B^2L^2 - RM\lambda} \ln\left( \frac{6B^2L^2}{RM\lambda} \right)$
D.
$\frac{3RM}{2B^2L^2 - 3RM\lambda} \ln\left( \frac{2B^2L^2}{3RM\lambda} \right)$

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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

Mass Defect vs. Binding Energy|Motional EMF in Rotating Rods|Lenz’s Law Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize modern physics (radioactive decay power) with electromagnetic induction and rigid body dynamics.

The primary trap lies in the misapplication of the Work-Energy Theorem: students often incorrectly assume a steady-state condition ( $P_{gen} = P_{diss}$ ) from the start, ignoring the inertial term ( $\frac{dK}{dt}$ ) which dictates the transient response.

Secondary traps involve incorrect formulations of the motional EMF for a rotating rod (forgetting the $1/2$ factor, leading to Option C) or using the wrong moment of inertia for a rod pivoted at one end (using $ML^2$ instead of $ML^2/3$ , leading to Option B).

The problem requires setting up and solving a first-order linear differential equation to find the exact time of maximum kinetic energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Mass Defect vs. Binding Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free