Discussion
If the ratio of the Young's modulii and densities of two rods of different materials are respectively, 3 : 2 and 3 : 1, then the ratio of the velocities of sound in the rods is
Answer: D
The ratio of the speeds of sound is 1 : \(\sqrt{2}\).
For longitudinal waves in a rod, \(v = \sqrt{\frac{Y}{\rho}}\):
\[ \begin{aligned} \frac{v_{1}}{v_{2}} \\ &= \sqrt{\frac{Y_{1}}{Y_{2}}\cdot\frac{\rho_{2}}{\rho_{1}}} \\ &= \sqrt{\frac{3}{2}\cdot\frac{1}{3}} \\ &= \frac{1}{\sqrt{2}} \end{aligned} \]
\(\sqrt{2} : 1\) inverts the density ratio, and 1 : 2 omits the square root.
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