An aluminum rod 1.6 m long is clamped at its centre. It is stroked with a rosin-coated cloth to set up a longitudinal vibration. The speed of sound in thin rod of aluminum is $5100 m s^{- 1}$ . The fundamental frequency of the waves established in the rod is $f_{0}$ . If the rod were copper (in which speed of sound is $3560 m s^{- 1}$ ), the fundamental frequency would have been $f_{0}^{1}$ . Then, the values of $f_{0}$ and $f_{0}^{1}$ are approximately,

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$f_{0} = 1594 H z$

$f_{0}^{1} = 1112 H z$

$f_{0} = 1690 H z$

$f_{0} = 1210 H z$

answer is A, B.

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Detailed Solution

As, the rod is centrally clamped, there is a node at the centre. In fundamental mode there are only the antinodes at the edges.
$\frac{λ_{0}}{2} = L \Rightarrow λ_{0} = 2 L f_{0} = \frac{v}{λ_{0}} = \frac{5000}{2 (1.6)} = 1590 H z$

In copper rod, $f_{0}^{1} = \frac{3560}{2 (1.6)} ≃ 110 H z$