Calculate the frequency to which piezoelectric oscillator circuitshould be tuned so that a piezoelectric crystal of thickness 0.1cmvibrates in its fundamental mode to generate ultrasonicwaves.(Young-s modulus and the density of material of crystalare 80 Gpa and 2654 kg/m3

The frequency of vibration of a piezoelectric crystal in its fundamental mode can be calculated using the formula:

f = v / 2d

where:

• f is the frequency,
• v is the speed of sound in the material, and
• d is the thickness of the crystal.

The speed of sound in a material can be calculated using the formula:

v = sqrt(E / ρ)

where:

• E is the Young's modulus of the material, and
• ρ is the density of the material.

Given that E = 80 GPa = 80 * 10^9 Pa and ρ = 2654 kg/m^3, we can calculate v as follows:

v = sqrt((80 * 10^9 Pa) / (2654 kg/m^3)) = 5483.3 m/s

Given that the thickness of the crystal d = 0.1 cm = 0.001 m, we can calculate f as follows:

f = v / 2d = 5483.3 m/s / (2 * 0.001 m) = 2.74 * 10^6 Hz

Therefore, the piezoelectric oscillator circuit should be tuned to a frequency of approximately 2.74 MHz for the piezoelectric crystal to vibrate in its fundamental mode and generate ultrasonic waves.