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Question #43496

Difficulty: 8/10 | 🟢 Free

The Question

A non-uniform heavy rope of length $L$ hangs vertically from a rigid support. The linear mass density (mass per unit length) of the rope varies as $\mu(x) = c x^2$ , where $x$ is the distance measured from its lower free end and $c$ is a positive constant. A small transverse wave pulse is generated at the lower end ( $x=0$ ). Find the time taken by the pulse to reach the rigid support at $x=L$ . Assume the acceleration due to gravity is $g$ and the rope is perfectly flexible.

Options

A.
$2\sqrt{\frac{3L}{g}}$
B.
$2\sqrt{\frac{L}{g}}$
C.
$2\sqrt{\frac{2L}{g}}$
D.
$\sqrt{\frac{3L}{g}}$

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

Tension in Massless vs Massive Strings|String Tension and Wave Speed|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model wave propagation in a medium with continuously varying properties.

A common trap is assuming the wave speed formula $v = \sqrt{T/\mu}$ implies a constant speed or incorrectly using the total mass to find an 'average' tension or density.

Students often mistakenly apply the uniform rope result ( $t = 2\sqrt{L/g}$ ) or fail to recognize that both tension and linear mass density are local functions of $x$ .

By setting $\mu(x) \propto x^2$ , the problem forces the explicit integration of the weight to find $T(x)$ and subsequently integrating $dt = dx/v(x)$ , rigorously testing the application of calculus in Newtonian mechanics and wave dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Tension in Massless vs Massive Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free