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 (\(\mu = 2\rho\)): \[ \frac{V_1}{V_2} = \sqrt{\frac{2\rho}{\rho}} = \sqrt{2}:1 \]