The speed of a longitudinal wave in a metallic bar is 400 ~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

Question

Quantrex Academy · Accepted Answer

Correct answer: B. 401

The speed of a longitudinal wave in a solid is given by

v = \sqrt{\frac{Y}{\rho}}

where

Y

is Young's modulus and

\rho

is density.
Initial condition:

v_0 = \sqrt{\frac{Y}{\rho}} = 400

m/s
After changes:

\rho' = 1.005\rho

(0.5% increase)

and Y' = 1.01Y

(1% increase)
New speed:

v' = \sqrt{\frac{Y'}{\rho'}} = \sqrt{\frac{1.01Y}{1.005\rho}} = \sqrt{\frac{1.01}{1.005}} \cdot \sqrt{\frac{Y}{\rho}}

\frac{v'}{v_0} = \sqrt{\frac{1.01}{1.005}} = \sqrt{1.00498...} \approx 1.00249

v' = 400 \times 1.00249 \approx 401

m/s

The speed of a longitudinal wave in a metallic bar is 400 ~m / s ....

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