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Question #42864

Difficulty: 8/10 | 🟢 Free

The Question

A string of length $L = 1 \text{ m}$ is kept under a constant tension $T = 100 \text{ N}$ . The linear mass density of the string is non-uniform and varies with the distance $x$ from one end as $\lambda(x) = \lambda_0(1 + \alpha x)$ , where $\lambda_0 = 0.01 \text{ kg/m}$ and $\alpha = 3 \text{ m}^{-1}$ . Using the approximation that the fundamental period of vibration is twice the time taken by a transverse wave pulse to travel the entire length of the string, determine the fundamental frequency of the string.

Options

A.
$\frac{225}{7} \text{ Hz}$
B.
$10\sqrt{10} \text{ Hz}$
C.
$\frac{100}{3} \text{ Hz}$
D.
$50 \text{ Hz}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Fundamental Frequency in Strings|SHM Definition via Differential Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle wave dynamics in non-homogeneous media.

A common trap is to incorrectly apply the standard formula $f = \frac{1}{2L}\sqrt{\frac{T}{\lambda}}$ by using the average linear mass density $\lambda_{avg}$ (leading to $10\sqrt{10} \text{ Hz}$ ) or by averaging the wave velocity (leading to $100/3 \text{ Hz}$ ).

Because the wave equation modifies to $T \frac{\partial^2 y}{\partial x^2} = \lambda(x) \frac{\partial^2 y}{\partial t^2}$ , the wave speed $v(x)$ varies continuously.

Students must integrate the infinitesimal time delay $dt = \frac{dx}{v(x)}$ to find the true wave transit time, applying the WKB-style approximation for the fundamental frequency.

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