Question

If percentage increase in Young's modulus \(Y\) is \(1%\), percentage increase in density of material is \(0.5%\) and longitudinal wave traveling in a metallic bar has wave velocity of \(400\,\text{m/s}\), then find final velocity of the wave.

• A. \(398\,\text{m/s}\)
• B. 355
• C. 401
• D. 402

Hint:

For wave speed relations involving square roots, always remember: \(\displaystyle \frac{\Delta v}{v} = \frac{1}{2}\frac{\Delta X}{X}\). Apply signs carefully when ratios are involved.

Answer 1

The Correct Option is C

Concept:
Velocity of a longitudinal wave in a solid rod is given by: \[ v = \sqrt{\frac{Y}{\rho}} \] Hence, fractional change in velocity is: \[ \frac{\Delta v}{v} = \frac{1}{2}\left(\frac{\Delta Y}{Y} - \frac{\Delta \rho}{\rho}\right) \]
Step 1: Substitute Percentage Changes
Given: \[ \frac{\Delta Y}{Y} = 1%, \quad \frac{\Delta \rho}{\rho} = 0.5% \] \[ \frac{\Delta v}{v} = \frac{1}{2}(1 - 0.5)% = \frac{1}{2}(0.5%) = 0.25% \]
Step 2: Calculate Final Velocity
Initial velocity: \[ v = 400\,\text{m/s} \] Increase in velocity: \[ \Delta v = 0.25% \times 400 = 1\,\text{m/s} \] \[ v_{\text{final}} = 400 + 1 = 401\,\text{m/s} \] \[ \boxed{v_{\text{final}} = 401\,\text{m/s}} \]