Question

The speed of the transverse waves on a steel wire of length 50 cm and mass 5 g subjected to a tension of 64 N is

• A. \(\sqrt{6.4} \text{ms}^{-1}\)
• B. \(8\sqrt{10} \text{ms}^{-1}\)
• C. \(80 \text{ms}^{-1}\)
• D. \(6.4 \text{ms}^{-1}\)
• E. \(640 \text{ms}^{-1}\)

Hint:

Double-check the units for \(\mu\). It must be kg/m. Forgetting to convert grams to kilograms is the most common error in string wave problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The speed of a transverse wave in a stretched string/wire depends on the tension in the wire and its mass per unit length (linear mass density).

Step 2: Key Formula or Approach:
Velocity \(v = \sqrt{\frac{T}{\mu}}\), where \(T\) is tension and \(\mu = \frac{m}{L}\).

Step 3: Detailed Explanation:
1. Convert units to SI:
Length \(L = 50 \text{ cm} = 0.5 \text{ m}\).
Mass \(m = 5 \text{ g} = 0.005 \text{ kg} = 5 \times 10^{-3} \text{ kg}\).
Tension \(T = 64 \text{ N}\).
2. Calculate linear mass density \(\mu\):
\[ \mu = \frac{m}{L} = \frac{0.005}{0.5} = 0.01 \text{ kg/m} \]
3. Calculate wave speed \(v\):
\[ v = \sqrt{\frac{64}{0.01}} = \sqrt{6400} \]
\[ v = 80 \text{ ms}^{-1} \]

Step 4: Final Answer:
The speed of the wave is \(80 \text{ ms}^{-1}\).