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Question #42477

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ is threaded on a fixed, smooth vertical circular wire of radius $R$ . The bead is projected from the lowest point of the wire with an initial speed $v_0$ . As it moves, it experiences a tangential air drag force of magnitude $F_d = \frac{m}{2R}v^2$ , where $v$ is its instantaneous speed. What is the minimum initial speed $v_0$ required for the bead to just reach the highest point of the circular wire?

Options

A.
$\sqrt{gR(e^\pi + 1)}$
B.
$\sqrt{gR(e^\pi - 1)}$
C.
$\sqrt{2gR(e^\pi + 1)}$
D.
$\sqrt{gR(e^{\pi/2} + 1)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Work-Energy Theorem Validity|Vertical Circular Motion|Non-Uniform Circular Motion

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in the presence of a path-dependent, non-conservative drag force.

Students commonly fall into the trap of attempting to use simple mechanical energy conservation ( $E_i = E_f$ ) or incorrectly integrating the drag force without coupling it to the kinematics.

By introducing a velocity-squared dependent drag, the problem forces the student to express the tangential acceleration as $v \frac{dv}{ds}$ and construct a first-order linear differential equation in $v^2$ with respect to the angular position $\theta$ .

This rigorously evaluates their ability to transform a path integral of a non-conservative force into a solvable angular differential equation, bridging advanced mechanics with calculus.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Work-Energy Theorem Validity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free