Question

If the speed of transverse waves on a stretched wire of linear density \(7\times 10^{-3}\,\text{kg m}^{-1}\) is \(100\,\text{m s}^{-1}\), then the tension in the wire is

• A. \(60\,\text{N}\)
• B. \(600\,\text{N}\)
• C. \(700\,\text{N}\)
• D. \(70\,\text{N}\)
• E. \(80\,\text{N}\)

Hint:

For waves on a string, always remember \(v=\sqrt{T/\mu}\). Rearranging gives \(T=\mu v^2\), which is very useful in numerical problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The speed of a transverse wave traveling on a stretched string or wire is determined by the tension in the wire and its linear mass density (mass per unit length).
Step 2: Key Formula or Approach:
The formula for the speed (\(v\)) of a transverse wave on a string is given by:
\[ v = \sqrt{\frac{T}{\mu}} \] where T is the tension in the string and \( \mu \) is the linear mass density. We need to rearrange this formula to solve for the tension, T.
Step 3: Detailed Explanation:
We are given:
- Speed of the wave, \( v = 100 \) ms\(^{-1}\).
- Linear density, \( \mu = 7 \times 10^{-3} \) kg m\(^{-1}\).
The formula is \( v = \sqrt{\frac{T}{\mu}} \).
To find the tension T, we first square both sides of the equation:
\[ v^2 = \frac{T}{\mu} \] Now, we can solve for T:
\[ T = v^2 \cdot \mu \] Substitute the given values into the equation:
\[ T = (100 \text{ ms}^{-1})^2 \cdot (7 \times 10^{-3} \text{ kg m}^{-1}) \] \[ T = (10^2)^2 \cdot 7 \times 10^{-3} \] \[ T = 10^4 \cdot 7 \times 10^{-3} \] \[ T = 7 \times 10^{4-3} \] \[ T = 7 \times 10^1 = 70 \text{ N} \] The calculated tension is 70 N. This matches option (D). The official answer key states "Question Cancelled", which might be due to an error in the provided key or some ambiguity in the question not apparent from the text. However, based on a straightforward calculation, the answer is 70 N.
Step 4: Final Answer:
The tension in the wire is 70 N. This corresponds to option (D).