Question

A string of density \(7.5\) g cm\(^{-3}\) and area of cross-section \(0.2\) mm\(^2\) is stretched under a tension of 20 N. When it is plucked at the mid-point, the speed of the transverse wave on the wire is

• A. 116 m s\(^{-1}\)
• B. 40 m s\(^{-1}\)
• C. 200 m s\(^{-1}\)
• D. 80 m s\(^{-1}\)

Hint:

\(v = \sqrt{T/\mu}\), where \(\mu = \rho A\) (density \(\times\) cross-sectional area). Always convert units to SI before substituting.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The speed of a transverse wave on a string is given by \(v = \sqrt{T/\mu}\), where \(T\) is tension and \(\mu\) is linear mass density.
Step 2: Detailed Explanation:
Linear mass density: \(\mu = \rho \times A = 7.5 \times 10^3 \text{ kg/m}^3 \times 0.2 \times 10^{-6} \text{ m}^2 = 1.5 \times 10^{-3}\) kg/m \[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{20}{1.5 \times 10^{-3}}} = \sqrt{13333} \approx 115.5 \approx 116 \text{ m/s} \]
Step 3: Final Answer:
The speed of the transverse wave is 116 m s\(^{-1}\).