Question

A string of length \(1\text{ m}\) and mass \(490\text{ g}\) is put under a tension of \(25\text{ N}\). A wave of frequency \(120\text{ Hz}\) is sent along it. The speed of this wave is

• A. \(7.14\text{ ms}^{-1}\)
• B. \(0.71\text{ ms}^{-1}\)
• C. \(0.51\text{ ms}^{-1}\)
• D. \(51.0\text{ ms}^{-1}\)

Hint:

For transverse waves on a stretched string, \[ v=\sqrt{\frac{T}{\mu}} \] where \(\mu=\frac{m}{L}\) is the mass per unit length.

Answer 1

The Correct Option is A

Step 1: Recall the formula for transverse wave speed on a string.
$v = \sqrt{T/\mu}$, where $T$ is tension and $\mu$ is linear mass density. The frequency (120 Hz) is irrelevant for wave speed; it only determines wavelength.
Step 2: Compute the linear mass density $\mu$.
\[ \mu = \frac{m}{L} = \frac{0.49 \text{ kg}}{1 \text{ m}} = 0.49 \text{ kg/m} \]
Step 3: Apply the wave speed formula.
$T = 25$ N: \[ v = \sqrt{\frac{25}{0.49}} = \sqrt{51.02} \approx 7.14 \text{ ms}^{-1} \]
Step 4: Explain why frequency is irrelevant.
Wave speed is a property of the medium (tension and density); frequency adjusts the wavelength via $\lambda = v/f$ but does not affect the speed.
Step 5: Check units.
$\sqrt{\frac{\text{N}}{\text{kg/m}}} = \sqrt{\text{m}^2\text{s}^{-2}} = \text{ms}^{-1}$. Correct.
Step 6: State the final answer.
\[ \boxed{v \approx 7.14 \text{ ms}^{-1}} \]