A steel rod $100\text{ cm}$ long is clamped at its middle. The fundamental frequency of longitudinal…
A steel rod $100\text{ cm}$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be $2.53\text{ kHz}$. What is the speed of sound in steel? [AIIMS 2018]
(a) $6.2\text{ kms}^{-1}$
(b) $5.06\text{ kms}^{-1}$
(c) $7.23\text{ kms}^{-1}$
(d) $7.45\text{ kms}^{-1}$
Solution
In fundamental mode, $l = 2 \left(\frac{\lambda}{4}\right) = \frac{\lambda}{2}$
[Diagram shows a standing wave pattern in a rod clamped at the center with a node $N$ at the center and antinodes $A$ at both ends over total length $l$]
$\Rightarrow \lambda = 2l$
Given, $l = 100\text{ cm} = 100 \times 10^{-2}\text{ m}, \nu = 2.53\text{ kHz} = 2.53 \times 10^3\text{ Hz}$
As, $v = \nu\lambda = \nu \times 2l$
$= 2.53 \times 10^3 \times 2 \times 100 \times 10^{-2}$
$= 5.06 \times 10^3\text{ ms}^{-1} = 5.06\text{ kms}^{-1}$