Question

Two vibrating strings of same length, same cross-sectional area and stretched to same tension are made of material with densities \(\rho\) and \(2\rho\). Each string is fixed at both ends. If \(V_1\) and \(V_2\) are speeds of transverse waves in the strings with densities \(\rho\) and \(2\rho\) respectively, then \(\frac{V_1}{V_2}\) is:

• A. \(1:2\)
• B. \(2:1\)
• C. \(\sqrt{2}:1\)
• D. \(1:\sqrt{2}\)

Hint:

When comparing the wave speeds in strings, remember that the speed is inversely proportional to the square root of the density, assuming the same tension and cross-sectional area.

Answer 1

The Correct Option is C

Step 1: Apply the formula for wave speed in a string. \[ V = \sqrt{\frac{T}{\mu}} \] where \(T\) is the tension and \(\mu\) is the linear mass density (\(\mu = \rho \times {area}\)). 
Step 2: Calculate the ratio of wave speeds. For string 1 (\(\mu = \rho\)) and string 2 (\(\mu = 2\rho\)): \[ \frac{V_1}{V_2} = \sqrt{\frac{2\rho}{\rho}} = \sqrt{2}:1 \]

Answer 2

To solve the problem, we need to determine the ratio of the speeds of transverse waves in two strings with different densities, given the same length, cross-sectional area, and tension.

1. Understanding the Relationship Between Wave Speed and Density:
The speed \( v \) of a wave on a string is given by the formula: \[ v = \sqrt{\frac{T}{\mu}} \] where: - \( T \) is the tension in the string, - \( \mu \) is the linear mass density of the string, which is given by \( \mu = \frac{m}{L} \), where \( m \) is the mass of the string and \( L \) is its length. For two strings with densities \( \rho \) and \( 2\rho \), the wave speeds \( v_1 \) and \( v_2 \) will be related by: \[ v_1 = \sqrt{\frac{T}{\rho}} \] \[ v_2 = \sqrt{\frac{T}{2\rho}} \] Thus, the ratio of the speeds of the two strings is: \[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{T}{\rho}}}{\sqrt{\frac{T}{2\rho}}} = \sqrt{\frac{2\rho}{\rho}} = \sqrt{2} \]

2. Identifying the Correct Answer:
The ratio \( \frac{v_1}{v_2} = \sqrt{2} \), which corresponds to Option 3.

Final Answer:
The correct answer is Option C: \( \sqrt{2} : 1 \).