Question

A steel rod $100\,\,cm$ long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be $2.53\,\,KHz$. What is the speed of sound in steel?

Answer 1

Hint: The speed of sound is explained as the distance travelled the sound per unit of time in the form of a wave, as it moves through an elastic that is a stretchable medium. The speed of sound resets strongly on temperature as well as the medium by which a sound wave is passing through.

Useful formula:
The speed of sound in steel is given by the relation:
\[s = f\lambda \]
Where, $s$ denotes the speed of the sound of the longitudinal vibrations of the steel rod, $f$ denotes the frequency of the longitudinal vibrations of the steel rod, $\lambda $ denotes the distance between the two successive nodes.

Complete step by step solution:
The data given in the problem are;
The length of the given steel rod is, $l = 100\,\,cm = 1\,m$
The fundamental frequency of the vibration is, $f = 2.53\,\,KHz$
The distance between the two successive nodes is $\dfrac{\lambda }{2}$.
The formula for the two successive node is;
$
  l = \dfrac{\lambda }{2} \\
  2l = \lambda \\
 $
Substitute the value of length of the steel rod $l = 1\,m$
$
  \lambda = 2 \times 1 \\
  \lambda = 2\,\,m \\
 $
The speed of sound in steel is given by the relation:
\[s = f\lambda \]
Where, $s$ denotes the speed of the sound of the longitudinal vibrations of the steel rod, $f$ denotes the frequency of the longitudinal vibrations of the steel rod.
\[
  s = 2.53 \times {10^3} \times 2 \\
  s = 5.06 \times {10^3}\,\,m\,s \\
  s = 5.06\,\,Km\,{s^{ - 1}} \\
 \]
Therefore, the speed of the longitudinal vibrations of the sound is given as \[s = 5.06\,\,Km\,{s^{ - 1}}\].

Note: A continuous cyclic change in the repositioning of elements of a rod-shaped object in the direction of the long axis of the rod is known as the longitudinal vibrations. The natural occurring frequency or the fundamental frequency can be referred to as that the lowest frequency of the repeated waveform.