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Question #43860

Difficulty: 8/10 | 🟢 Free

The Question

A uniform elastic string of length $L$ , uniform cross-sectional area $A$ , and volume mass density $\rho$ is rotated in a horizontal plane about one of its ends with a constant angular velocity $\omega$ . The breaking stress of the material of the string is $\sigma_b$ . Neglecting gravity, determine the maximum angular velocity $\omega_{max}$ the string can withstand without breaking, and the speed $v$ of a transverse wave pulse at the midpoint of the string ( $r = L/2$ ) when it is rotating at $\omega_{max}$ .

Options

A.
$\omega_{max} = \frac{1}{L}\sqrt{\frac{2\sigma_b}{\rho}}, \quad v = \sqrt{\frac{3\sigma_b}{4\rho}}$
B.
$\omega_{max} = \frac{1}{L}\sqrt{\frac{2\sigma_b}{\rho}}, \quad v = \sqrt{\frac{\sigma_b}{\rho}}$
C.
$\omega_{max} = \frac{1}{L}\sqrt{\frac{\sigma_b}{\rho}}, \quad v = \sqrt{\frac{\sigma_b}{2\rho}}$
D.
$\omega_{max} = \frac{1}{L}\sqrt{\frac{\sigma_b}{\rho}}, \quad v = \sqrt{\frac{\sigma_b}{\rho}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

String Tension and Wave Speed|Breaking Stress Concept|Stress vs. Strain Definition

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-uniform stress distribution in a rotating body and its effect on wave propagation.

The primary trap is treating the string as a point mass at the center of mass or the end, leading to an incorrect uniform tension profile $T = M\omega^2 L$ or $T = M\omega^2 L/2$ .

Students falling for this will incorrectly calculate $\omega_{max}$ and assume a constant wave speed $v = \sqrt{T/\mu}$ throughout the string, ignoring the radial dependence of tension $T(r)$ required by the local centripetal force gradient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	String Tension and Wave Speed
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free