Question

A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is 2.7×103 kgm-3 and its Young's modulus is 9.27×1010 Pa. What will be the fundamental frequency of the longitudinal vibrations?
(A) 5 kHz
(B) 2.5 kHz
(C) 10 kHz

Answer 1

Hint
In order to find the fundamental frequency, we use the formula $\upsilon = \dfrac{v}{\lambda }$, v is the velocity of the wave. We can find the velocity by using the formula $v = \sqrt {\dfrac{\gamma }{\rho }} $, γ is the young’s modulus and ρ is the density. As, the length is clamped at its middle point so wavelength $\lambda = 2L$
By using all these formulas, we will get the desired result.

Complete step by step answer
It is given that, Density of granite is $\rho = 2.7 \times {10^3}kg{m^{ - 3}}$
Young’s modulus of graphite is $\gamma = 9.27 \times {10^{10}}Pa$
The velocity of the longitudinal vibration is $v = \sqrt {\dfrac{\gamma }{\rho }} $
γ is the young’s modulus and ρ is the density
on substituting the values, we get
$ \Rightarrow v = \sqrt {\dfrac{{9.27 \times {{10}^{10}}}}{{2.7 \times {{10}^3}}}} $
$ \Rightarrow v = \sqrt {3.433 \times {{10}^7}} = \sqrt {34.33} \times {10^3}$
$ \Rightarrow v = 5.85 \times {10^3}m{s^{ - 1}}$
Now, as the granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations.
Therefore, the half of the wavelength is equal to the length of the graphite i.e. $\lambda = 2L$
As length of graphite is 60cm then the wavelength is equal to the
$ \Rightarrow \lambda = 2 \times 60cm = 120cm = 1.2m$
As, the relation between frequency, wavelength and velocity is $\upsilon = \dfrac{v}{\lambda }$
Where, ν is the frequency of vibration
V is the velocity of the vibration.
On substituting the values, we get
$ \Rightarrow \nu = \dfrac{{5.85 \times {{10}^3}}}{{1.2}} = 4.88 \times {10^3}Hz$
$ \Rightarrow v = 5kHz$
Hence, the fundamental frequency of the longitudinal vibration is 5kHz.
Thus, option (A) is correct.

Note
There are two types of vibrations, one is longitudinal and other is transverse vibration that depends upon the direction in which the wave is traveling. If the wave shakes in the same direction of the direction of motion then this is known as longitudinal wave. And if the wave shaken is in a perpendicular direction to the direction of motion that wave is called transverse wave.