An aluminium rod having a length $100 \,cm$ is clamped at its middle point and set into longitudinal vibrations. Let the rod vibrate in its fundamental mode. The density of aluminium is $2600 \,kg / m ^3$ and its Young's modulus is $7.8 \times 10^{10} \,N / m ^2$. The frequency of the sound produced is .............. $Hz$
Medium
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(b)
$v=\sqrt{\frac{y}{\rho}}=\sqrt{\frac{7.8 \times 10^{10}}{2600}}=5480 \,ml$
Since rod is clamped at the middle, the middle point is a pressure antinode and free ends are nodes. In the fundamental mode there are no other nodes and antinodes. The length of the rod is therefore half the wavelength.
So, $\lambda=2 I=2 \,m$
Frequency $=\frac{v}{\lambda}=\frac{5480}{2}=2740 \,Hz$
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