Before You Begin

Question #42402

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is attached to a light inextensible string of length $R$ and suspended from a fixed point. It is projected horizontally from the lowest point with an initial speed $v_0$ . During its motion in the vertical plane, it experiences a non-conservative tangential drag force $F_d = - \frac{k m}{2R} v^2$ , where $v$ is its instantaneous speed and $k$ is a positive dimensionless constant. By recognizing the isomorphism of its kinetic energy evolution to a driven dissipative circuit, determine the minimum initial speed $v_0$ required for the particle to just complete the full vertical circle.

Options

A.
$\sqrt{ gR \left( \frac{k^2+3}{k^2+1} e^{k\pi} + \frac{2}{k^2+1} \right) }$
B.
$\sqrt{ 2gR \left( \frac{e^{k\pi} + 1}{k^2+1} \right) }$
C.
$\sqrt{ gR \left( 5 e^{k\pi} - \frac{2k}{k^2+1} \right) }$
D.
$\sqrt{ gR \left( \frac{k^2+3}{k^2+1} e^{-k\pi} + \frac{2}{k^2+1} \right) }$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Minimum Speed for Looping|L-C Circuit Analogy|Energy Partitioning|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-conservative forces in a vertical loop by mapping the energy decay to a driven dissipative circuit (L-R analogue).

The primary trap is the 'Energy Partitioning' misconception, where students either incorrectly assume energy conservation (yielding the classical $\sqrt{5gR}$ ) or fail to apply the correct boundary condition at the top ( $v_{top} \ge \sqrt{gR}$ ), mistakenly assuming $v_{top} = 0$ (Option B).

By forcing the integration of a time-varying potential (gravity as a function of angle) with a velocity-dependent drag, it requires treating the system as a first-order linear ODE, shifting the minimum speed boundary condition to a transient exponential solution.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Minimum Speed for Looping
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free