Question
The speed of a longitudinal wave in a metallic bar is $400\text{ }m/s$. If the density and Young's modulus of the bar material are increased by $0.5\%$ and $1\%$ respectively then the speed of the wave is changed approximately to $\_\_\_\_$ $m/s$.
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Solution
$V_{\text{sound~}} = \sqrt{\frac{Y}{\rho}}$
$$\begin{matrix} & \frac{\Delta V}{V} \times 100 = \frac{1}{2}\left( \frac{\Delta Y}{Y} \times 100 \right) - \frac{1}{2}\left( \frac{\Delta\rho}{\rho} \times 100 \right) \\ & \ = \frac{1}{2} \times 1 - \frac{1}{2} \times 0.5 \\ & \frac{\Delta V}{V} \times 100 = \frac{1}{4} \\ & \Delta V = \frac{1}{4} \times \frac{V}{100} \\ & \Delta V = 1\text{ }m/s \\ & V_{\text{final~}} = 400 + 1 = 401\text{ }m/s \end{matrix}$$
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