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Question #42088

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is moving on a smooth horizontal table, attached to a string passing through a small hole in the table. Initially, the particle moves in a circle of radius $R$ with speed $v_0$ . At $t=0$ , the string is pulled downwards such that the radial distance of the particle from the hole decreases at a constant rate $u = \frac{2v_0}{\pi}$ . What is the magnitude of the change in the velocity vector of the particle from the instant just after the pulling begins until its radial distance becomes $R/2$ ?

Options

A.
$v_0 \sqrt{5 - \frac{4}{\pi} + \frac{8}{\pi^2}}$
B.
$v_0 \left( \sqrt{4 + \frac{4}{\pi^2}} - \sqrt{1 + \frac{4}{\pi^2}} \right)$
C.
$v_0 \sqrt{5}$
D.
$v_0 \sqrt{9 + \frac{16}{\pi^2}}$

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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

Change in Velocity (Δv)|Angular Momentum conservation|System of Particles and Rotational Motion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the vector nature of velocity in central force motion and the application of angular momentum conservation.

A common trap is calculating the scalar difference in speeds (Option B) or incorrectly assuming the radial and tangential unit vectors remain constant in direction, leading to a simple Cartesian subtraction without accounting for the rotation of the basis vectors.

To solve it correctly, students must recognize that the non-constant angular velocity must be integrated over the time-dependent radius to find the exact angular displacement ( $\Delta \theta = \pi/2$ ), which dictates the transformation of the local polar coordinate basis vectors before the initial and final velocity vectors can be subtracted.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Change in Velocity (Δv)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free