Question

A piezo-electric quartz plate is vibrating to produce a wavelength of ${{10}^{-2}}m$. Find the frequency if for quartz $Y=8\times {{10}^{10}}N{{m}^{-2}}$ and $\rho =2.65\times {{10}^{3}}kg{{m}^{-3}}$
A. $350kHz$
B. $450kHz$
C. $550kHz$
D. $650kHz$

Answer 1

Hint: Try to find out the velocity with the given data by using the relation between the young’s modulus and velocity, using this velocity finds the required frequency.

Complete step by step answer:
The given data is as follows
Wavelength of the vibrating plate is $\lambda =0.01m$
Young’s modulus $Y=8\times {{10}^{10}}N{{m}^{-2}}$
Density of the plate is $\rho =2.65\times {{10}^{3}}kg{{m}^{-3}}$
We know the relation between the velocity and young’s modulus as
$Y={{v}^{2}}\rho $
Where $Y=\text{young }\!\!'\!\!\text{ s modulus}$
$v=\text{velocity}$
$\Rightarrow {{v}^{2}}=\dfrac{Y}{\rho }$
$\Rightarrow v=\sqrt{\dfrac{Y}{\rho }}$
Substitute the above values in this formula and calculate the velocity
$\Rightarrow v=\sqrt{\dfrac{Y}{\rho }}$
$\Rightarrow v=\sqrt{\dfrac{8\times {{10}^{10}}}{2.65\times {{10}^{3}}}}$
$\Rightarrow v=5494.4m{{s}^{-1}}$
$\therefore v=5.5\times {{10}^{3}}m{{s}^{-1}}$
Therefore the required velocity is $v=5.5\times {{10}^{3}}m{{s}^{-1}}$
We also have
We know that frequency $f=\dfrac{v}{\lambda }$
Where $f=\text{frequency}$
$v=\text{velocity}$
$\lambda =\text{wavelength}$
Now substitute the above given values in this formula to find the required frequency of the vibrating plate
$\Rightarrow f=\dfrac{v}{\lambda }$
$\Rightarrow f=\dfrac{5.5\times {{10}^{3}}}{{{10}^{-2}}}$
$\therefore f=55\times {{10}^{4}}=550kHz$
Therefore, the required value of the frequency of the vibrating plate is $f=550kHz$
So, the correct answer is “Option C”.

Note:
While solving a problem if we are not having the required data exactly and instead they have given some other terms then we have to check that whether we can get the required terms with the extra terms given i.e., indirectly given the required value which we have to find by a relation with the given terms and then we have to proceed to find the required parameters.