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Question #43489

Difficulty: 8/10 | 🟢 Free

The Question

A small particle of mass $m$ is placed on a smooth horizontal table. It is attached to one end of a light, inextensible, and initially slack string of length $L = \sqrt{5}R$ . The other end of the string is fixed at the origin $O(0,0)$ . At $t=0$ , the particle is at point $A(0, R)$ and is projected with a velocity $v_0 \hat{i}$ . During its motion, the particle experiences a resistive drag force $\vec{F} = -m\alpha v \vec{v}$ , where $\alpha$ is a positive constant, $v$ is its instantaneous speed, and $\vec{v}$ is its velocity vector. Find the tension in the string immediately after it becomes taut.

Options

A.
$\frac{m v_0^2 e^{-4\alpha R}}{\sqrt{5}R}$
B.
$\frac{m v_0^2 e^{-2\alpha R}}{5\sqrt{5}R}$
C.
$\frac{m v_0^2 e^{-4\alpha R}}{5\sqrt{5}R}$
D.
$\frac{4m v_0^2 e^{-4\alpha R}}{5\sqrt{5}R}$

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

Non-Uniform Circular Motion|Non-Uniform Acceleration Constraint|Calculus Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of non-uniform kinematics with impulsive constraints and circular dynamics.

The primary trap lies in the impulsive phase: students often incorrectly assume that kinetic energy is conserved when a string becomes taut, failing to realize that the radial component of velocity is permanently destroyed by the inelastic jerk (Trap A).

Another common mistake is incorrectly projecting the velocity by misidentifying the angle between the velocity vector and the string, leading to Trap D.

Furthermore, the non-uniform acceleration requires integrating $v dv/dx = -\alpha v^2$ ; students who misapply constant acceleration formulas or make algebraic errors in squaring the exponential will fall into Trap B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Non-Uniform Circular Motion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free