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Question #42447

Difficulty: 8/10 | 🟢 Free

The Question

A uniform thick rope of mass $M$ , length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is placed on a smooth horizontal surface. A constant horizontal force $F$ is applied at one end to accelerate it along its length. Let $\Delta L$ be the total elongation of the rope in steady state (assuming $\Delta L \ll L$ ), and $t_0$ be the time taken by a small transverse wave pulse to travel from the free end to the pulled end. Which of the following expressions represents the ratio $\frac{\Delta L}{t_0^2}$ ?

Options

A.
$\frac{F^2}{8 M A Y}$
B.
$\frac{F^2}{4 M A Y}$
C.
$\frac{F^2}{2 M A Y}$
D.
$\frac{F^2}{M A Y}$

Check Your Answer

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

Newton’s Third Law Pairing|Young's Modulus Sensitivity|Stress vs. Strain Definition

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle continuous mass distributions under acceleration, specifically targeting the misconception of 'Uniform Tension'.

Students often incorrectly assume that a pulling force creates a uniform stress $F/A$ across the entire rope, leading to an incorrect elongation $\Delta L = FL/AY$ .

Furthermore, it integrates Wave Dynamics by requiring the calculation of wave speed in a medium with a spatial tension gradient $T(x)$ .

The trap lies in failing to set up the differential elements for both the elastic strain $d(\Delta x) = T(x)dx/(AY)$ and the wave transit time $dt = dx/\sqrt{T(x)/\mu}$ , thereby testing true mastery of calculus-based physics over rote formula memorization.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Newton’s Third Law Pairing
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free