Before You Begin

Question #42465

Difficulty: 8/10 | 🟢 Free

The Question

A uniform heavy string of mass $M$ and length $L$ is placed on a smooth horizontal table. A small particle of mass $m$ is attached to one end of the string. The other end of the string is passed through a small smooth hole in the table. Initially, the string is fully extended on the table and the system rotates about the hole with an angular velocity $\omega_0$ . The string is then pulled slowly from below until the length of the string remaining on the table is $L/2$ . At this instant, a small transverse pulse is generated at the hole. Find the time taken by the pulse to reach the particle.

Options

A.
$\frac{\sqrt{2} (M + 6m)}{8 \omega_0 (M + 3m)} \arcsin\left( \sqrt{\frac{M}{M + 4m}} \right)$
B.
$\frac{\sqrt{2} (M + 3m)}{4 \omega_0 (M + 6m)} \arcsin\left( \sqrt{\frac{M}{M + 2m}} \right)$
C.
$\frac{M + 6m}{4 \omega_0 (M + 3m)} \sqrt{\frac{M}{m}}$
D.
$\frac{\sqrt{2} (M + 4m)}{8 \omega_0 (M + 2m)} \ln\left( 1 + \frac{M}{4m} \right)$

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

Recoil and Thrust Forces|Tension in Massless vs Massive Strings|SHM Definition via Differential Equation

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of rigid body dynamics and wave mechanics in continuous media.

Students often fall into the trap of treating the string as massless when calculating tension, or they use a lumped-mass approximation, assuming constant tension $T = m \omega^2 r$ .

By doing so, they fail to account for the string's own inertia contributing to the local centripetal force, which makes the tension—and consequently the wave speed—a function of the radial coordinate $x$ .

Furthermore, the problem requires careful application of angular momentum conservation; as the massive string is pulled through the hole, the rotating mass distribution changes, altering the system's moment of inertia.

Solving this requires setting up a differential equation for the wave propagation time $dt = dx/v(x)$ and integrating it over the new boundary conditions, effectively transitioning from simple particle mechanics to distributed parameter wave dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Recoil and Thrust Forces
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free