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Question #42865

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $M$ is constrained to move on a frictionless horizontal track. A string of uniform linear mass density $\mu$ is attached to the block, passes through a small smooth ring fixed at a height $h$ directly above the track, and supports a hanging mass $m_0$ . An external mechanism maintains the block's velocity at a constant value $v$ directed away from the point directly below the ring. At the instant the block is at a horizontal distance $x$ from the point directly below the ring, what is the instantaneous fundamental frequency of the transverse vibrations of the string segment between the ring and the block? (Define the frequency as the inverse of the round-trip time of a wave pulse, and assume $v$ is sufficiently small).

Options

A.
$\frac{1}{2\sqrt{x^2+h^2}} \sqrt{\frac{m_0 g}{\mu}}$
B.
$\frac{1}{2\sqrt{x^2+h^2}} \left( \sqrt{\frac{m_0}{\mu} \left( g + \frac{h^2 v^2}{(x^2+h^2)^{3/2}} \right)} - \frac{x v}{\sqrt{x^2+h^2}} \right)$
C.
$\frac{1}{2\sqrt{x^2+h^2}} \sqrt{\frac{m_0}{\mu} \left( g + \frac{h^2 v^2}{(x^2+h^2)^{3/2}} \right)}$
D.
$\frac{1}{2\sqrt{x^2+h^2}} \left( \sqrt{\frac{m_0 g}{\mu}} - \frac{x v}{\sqrt{x^2+h^2}} \right)$

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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

Fundamental Frequency in Strings|Constraint Relations via String Length|Work-Energy Theorem Validity

Why This Question Is Tricky

Common Mistakes

This question tests two advanced concepts: dynamic tension via constraint relations and the geometric Doppler shift of wave frequencies.

Students commonly misapply Newton's laws by assuming the tension is simply $T = m_0 g$ , ignoring the upward acceleration of the hanging mass caused by the nonlinear constraint $L = \sqrt{x^2+h^2}$ .

Furthermore, students often blindly apply the static frequency formula $f = c/(2L)$ .

Because the string is drawn through the ring, the medium moves with velocity $\dot{L}$ , requiring a modified round-trip time calculation that introduces a first-order Doppler shift $f = (c-\dot{L})/(2L)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Fundamental Frequency in Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free