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Question #43359

Difficulty: 8/10 | 🟢 Free

The Question

A small conducting bead of mass $m$ is threaded on a fixed, rough, horizontal circular wire of radius $r$ . The wire has a uniform resistance with a total resistance $R_0$ . A constant current $I$ is maintained in the circular wire by an external source connected across a negligible gap at $\theta = 0$ . The bead acts as a wiper for a potentiometer circuit: an ideal cell of electromotive force $\varepsilon$ and a galvanometer are connected in series between the gap at $\theta = 0$ and the bead. The coefficient of kinetic friction between the bead and the wire is $\mu$ . The bead is projected from $\theta = 0$ along the wire with an initial tangential speed $v_0$ . It is observed that the bead comes to rest exactly at the angular position where the galvanometer shows zero deflection (the null point). Assuming the normal force on the bead is the vector sum of the vertical component balancing gravity and the horizontal radial component providing centripetal acceleration, find the initial speed $v_0$ . (Acceleration due to gravity is $g$ ).

Options

A.
$\sqrt{rg \sinh \left( \frac{4\pi \mu \varepsilon}{I R_0} \right)}$
B.
$\sqrt{ \frac{4\pi \mu r g \varepsilon}{I R_0} }$
C.
$\sqrt{rg \cosh \left( \frac{4\pi \mu \varepsilon}{I R_0} \right)}$
D.
$\sqrt{rg \sinh \left( \frac{2\pi \mu \varepsilon}{I R_0} \right)}$

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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

Potentiometer Sensitivity|Non-Uniform Circular Motion|Kinetic Friction Transition

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem and Newton's Second Law in a non-uniform circular motion scenario where the non-conservative frictional force depends continuously on the transient velocity.

A common trap is failing to recognize that the normal force is the vector sum of the vertical gravitational component and the horizontal centripetal component, leading students to incorrectly assume a constant frictional force $f_k = \mu mg$ .

This results in a simple linear deceleration and leads to the incorrect Option B.

Another trap is incorrectly integrating the resulting non-linear differential equation (missing the factor of 2) or failing to map the mechanical stopping condition to the electrical null-point constraint of the potentiometer circuit.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Potentiometer Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free