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Question #43490

Difficulty: 8/10 | 🟢 Free

The Question

A stationary bob of mass $m$ is suspended from a fixed pivot by a massless string of length $L$ . A small bullet of mass $m$ moving horizontally with speed $u = 2\sqrt{5gL}$ strikes the bob and gets embedded in it. The string has a variable un-stretched cross-sectional area given by $A(\theta) = A_0 (2 + \cos\theta)^2$ , where $\theta$ is the angular displacement of the string from the lowest point and $A_0$ is a positive constant. If the string is designed such that it just reaches its breaking stress $\sigma_b$ during the subsequent motion, find the required value of $\sigma_b$ and the angle $\theta_b$ at which the string breaks.

Options

A.
$\sigma_b = \frac{3mg}{2A_0}, \; \theta_b = \frac{\pi}{2}$
B.
$\sigma_b = \frac{4mg}{3A_0}, \; \theta_b = 0$
C.
$\sigma_b = \frac{36mg}{25A_0}, \; \theta_b = \frac{\pi}{3}$
D.
$\sigma_b = \frac{3mg}{4A_0}, \; \theta_b = \frac{\pi}{2}$

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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

Non-Uniform Circular Motion|Breaking Stress Concept|Calculus of Variations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of collision mechanics, non-uniform circular motion, and material stress.

A common trap (tested by Option B) is assuming that the string will break at the lowest point ( $\theta = 0$ ) simply because the tension is maximum there.

However, stress is the ratio of tension to cross-sectional area.

Because the area $A(\theta)$ decreases as $\theta$ increases, students must express stress as a continuous function of $\theta$ and use calculus to find its true maximum.

Additionally, students must correctly apply conservation of angular momentum during the perfectly inelastic collision to find the correct initial velocity, and remember to use the combined mass $2m$ for the subsequent dynamics (Option D traps those who forget the mass addition).

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