Understanding Piezoelectric Transducer Force Measurement

A piezoelectric transducer converts mechanical stress or force into an electrical charge or voltage. This property is utilized to measure applied forces. The amount of voltage generated is directly proportional to the applied force, mediated by the transducer's material properties and geometry.

Key Parameters for Piezoelectric Force Calculation

In this problem, we are given the following information:

• The size of the piezoelectric crystal: 6 mm × 6 mm × 2 mm.
• The voltage sensitivity ($g$) of the crystal: 0.055 Vm/N. This value represents the piezoelectric voltage coefficient.
• The developed voltage ($V$): 120 V.

We need to find the force ($F$) applied to the transducer.

Formula for Voltage Development in Piezoelectric Transducers

The voltage ($V$) developed across a piezoelectric crystal due to an applied force ($F$) is related by the following formula, considering the piezoelectric voltage coefficient ($g$), the area ($A$) over which the force is applied, and the thickness ($t$) of the crystal:

The relationship is given by:

$$ V = g \cdot \frac{F}{A} \cdot t $$

Where:

• $V$ is the developed voltage (in Volts, V).
• $g$ is the piezoelectric voltage coefficient (in Volt-meter per Newton, Vm/N).
• $F$ is the applied force (in Newtons, N).
• $A$ is the area of the crystal faces perpendicular to the force (in square meters, m²).
• $t$ is the thickness of the crystal in the direction of voltage measurement (in meters, m).

Step-by-Step Calculation of Applied Force

First, let's list the given values and convert them into standard SI units (meters, Newtons, Volts):

• Voltage sensitivity, $g = 0.055 \text{ Vm/N}$
• Developed voltage, $V = 120 \text{ V}$
• Crystal dimensions: Length = 6 mm, Width = 6 mm, Thickness = 2 mm.

Now, calculate the area ($A$) and thickness ($t$) in square meters and meters, respectively:

• Area, $A = \text{Length} \times \text{Width} = 6 \text{ mm} \times 6 \text{ mm} = 36 \text{ mm}^2$
• Convert Area to m²: $A = 36 \times (10^{-3} \text{ m})^2 = 36 \times 10^{-6} \text{ m}^2$
• Thickness, $t = 2 \text{ mm}$
• Convert Thickness to m: $t = 2 \times 10^{-3} \text{ m}$

Next, rearrange the formula $ V = g \cdot \frac{F}{A} \cdot t $ to solve for the force ($F$):

$$ F = \frac{V \cdot A}{g \cdot t} $$

Substitute the values into the rearranged formula:

$$ F = \frac{120 \text{ V} \times (36 \times 10^{-6} \text{ m}^2)}{(0.055 \text{ Vm/N}) \times (2 \times 10^{-3} \text{ m})} $$

Perform the calculation:

$$ F = \frac{120 \times 36 \times 10^{-6}}{0.055 \times 2 \times 10^{-3}} \text{ N} $$

$$ F = \frac{4320 \times 10^{-6}}{0.11 \times 10^{-3}} \text{ N} $$

$$ F = \frac{4320}{0.11 \times 10^{(-3 - (-6))}} \text{ N} $$

$$ F = \frac{4320}{0.11 \times 10^3} \text{ N} $$

$$ F = \frac{4320}{110} \text{ N} $$

$$ F = \frac{432}{11} \text{ N} $$

$$ F \approx 39.2727... \text{ N} $$

Result and Verification

The calculated force is approximately 39.27 N. Comparing this result with the given options, we find that option 3 (39.24 N) is the closest value. The slight difference is likely due to rounding in the provided sensitivity value or the options themselves.