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

Given, the length of the steel rod is L=100 cmand the fundamental frequency of the longitudinal vibrations of the rod is 2.53 kHz.

Since, the steel rod is clamped at the middle, so the fundamental node of vibration of the rod is formed such that it has a node in the middle and an anti node at each end as shown in the figure below.

The formula to calculate the length of the steel rod is,

L= λ 4 + λ 4 = λ 2 λ=2L

Here, the length of the rod is L and the wavelength of the vibration is λ.

Substituting the value in the above equation, we get:

λ=2( 100 ) =200× 1 cm 100 m =2 m

The formula to calculate the speed of sound in steel is,

u=vλ

Here, the fundamental frequency of the longitudinal vibrations of the rod is v.

Substituting the value in the above equation, we get:

u=( 2.53× 10 3  Hz 1 kHz )( 2 m ) =( 2.53× 10 3 )( 2 ) =5.06× 10 3  m/s =5.06 km-s −1

Thus, the speed of sound in steel is 5.06 km-s −1 .