Question

A piezoelectric quartz crystal of thickness $0.005\mathrm{m}$ is vibrating in resonance conditions. Calculate the fundamental frequency ${\mathrm{f}}_0$ for quartz.

$\left(\mathrm{\gamma }=8\times {10}^{10}{\mathrm{Nm}}^{-2}\mathrm{and}\mathrm{\rho }=2.65\times {10}^3{\mathrm{kg}}^{-3}\right)$ .

• A. $55\mathrm{MHz}$
• B. $0.55\mathrm{MHz}$
• C. $5.5\mathrm{MHz}$
• D. $5.5\mathrm{kHz}$

Answer 1

The correct option is B

$0.55\mathrm{MHz}$

Step 1. Given data

It is given that a thickness of piezoelectric quartz crystal is $t=0.005\mathrm{m}$,

$\mathrm{\gamma }=8\times {10}^{10}{\mathrm{Nm}}^{-2}$

$\mathrm{\rho }=2.65\times {10}^3{\mathrm{kg}}^{-3}$

Where, $\mathrm{\gamma }$ is the young's modulus and $\mathrm{\rho }$ is the density of material.

The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency or base frequency of a periodic waveform.

It is calculated by,

${\mathrm{V}}_0=\frac{1}{2\mathrm{L}}\times \mathrm{v}$

Here, $\mathrm{L}$ is the length or thickness, and ${\mathrm{V}}_0$ is the fundamental frequency.

Step 2. Find the velocity of wave.

We know that, velocity of wave is,

$\mathrm{\nu }=\sqrt{\frac{\mathrm{\gamma }}{\mathrm{\rho }}}$

Here, $\mathrm{v}$ is the velocity of wave, $\mathrm{\gamma }$ is the young's modulus and $\mathrm{\rho }$ is the density of material.

Substituting the given values, we get,

$\mathrm{\nu }=\sqrt{\frac{8\times {10}^{10}}{2.65\times {10}^3}}$

$=\sqrt{30188679.25}$

$=5494.422558$

$=5.5\times {10}^3\mathrm{m}/\sec$

For the fundamental mode of vibration the thickness should be equal to $\frac{\mathrm{\lambda }}{2}$.

So,

$\mathrm{\lambda }=2\mathrm{t}$

Here, $\mathrm{t}$ is thickness.

$\mathrm{\lambda }=2\times 0.005$

$=0.01\mathrm{meter}$

Step 3. Calculate the fundamental frequency.

Now, the wavelength, frequency and velocity are related as,

$\mathrm{\nu }=\mathrm{n\lambda }$

or

$\mathrm{n}=\frac{\mathrm{v}}{\mathrm{\lambda }}$

$=\frac{5.5\times {10}^3}{0.01}$

$=550000$

$=5.5\times {10}^5\mathrm{Hz}$

Now, we will convert the above calculated value from $\mathrm{Hz}$ to $\mathrm{MHz}$. we get,

$\mathrm{MHz}=\frac{\mathrm{Hz}}{1000000}$

$=\frac{5.5\times {10}^5}{1000000}$

$=0.55$

So, the fundamental frequency ${\mathrm{f}}_0$ for quartz is $0.55\mathrm{MHz}$.

Hence, option $\mathrm{B}$ is correct answer.