A Copper Metal Wire Is Used As A Strain Gauge The Resistivit
A Copper Metal Wire Is Used As A Strain Gauge The Resistivity Is 168
A copper metal wire is used as a strain gauge. The resistivity is 1.68×10⁻⁸ Ω·m at 20°C. The length and cross-sectional area of the wire are 5 mm and 4×10⁻⁴ m², respectively. The material elongates by an amount of 0.2 mm in 0.2 mm increments until it reaches 6 mm in length. Assuming the volume remains constant, calculate the resistance at each length. You should use the standard equation for the resistance of a metal. What is that equation? Fill in the table below. Show all calculations. Calculate the difference in resistance between the resistance at each length and the resistance before strain is added. Also, calculate the change in resistance using the approximation found in equation 5.12 of your text. How does the change in length change the resistance of the gauge? Is it linear? Why or why not? (You can use Excel to create a plot and paste it in your submission if you want.) How good is the approximation? Length Cross-sectional area Resistance Initial resistance (at 5 mm) Change in the resistance Change in resistance using Eq. 5.12 5 mm 5.2 mm 5.4 mm 5.6 mm 5.8 mm 6 mm A tensile force of 2150 N is applied to a 12 m steel beam with a cross-sectional area of 5.2×10⁻⁴ m².
Find the strain on the beam. A strain gauge has a GF (Gauge factor) = 2.03 and R = 110 ohms and is made from wire with α = 0.0035/°C at 25°C. The dissipation factor is given as PD = 20 mW. What is the maximum current that can be placed through the strain gauge to keep self-heating errors below 1% of strain? The atmospheric pressure is 14.5 psi.
If the absolute pressure is 2,865.6 psfa, what is the gauge pressure? What is the gauge pressure in (a) kPa, and (b) N/cm², at a distance 5.5 ft below the surface of a column of water?
What is the flow rate in liters per second through a pipe 32 cm in diameter, if the average velocity is 2.1 m/s?
A conveyor belt is traveling at 27 cm/s, and a load cell with a length of 0.72 m is reading 5.4 kg. What is the flow rate of the material on the belt?
Paper For Above instruction
Introduction
The application of strain gauges using metallic wires, particularly copper, is a fundamental method in mechanical engineering for the measurement of strain. Understanding the resistance changes due to strain involves core electrical principles and material properties. This paper examines the resistance behavior of a copper wire used as a strain gauge under elongation, assesses the effects of applied tensile force on a steel beam, evaluates pressure calculations in fluid columns, and determines flow and material transport rates in piping and conveyor systems. The comprehensive analysis combines theoretical equations, the application of material properties, and practical engineering calculations, facilitating accurate measurement and control in various engineering contexts.
Resistance Calculation of Copper Wire under Strain
The resistance \( R \) of a metallic wire is determined by the fundamental relation:
\[
R = \frac{\rho L}{A}
\]
where:
- \( \rho \) is the resistivity,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.
Given the resistivity \( \rho = 1.68 \times 10^{-8} \, \Omega \cdot \text{m} \), initial length \( L_0 = 5\, \text{mm} = 0.005\,\text{m} \), and area \( A = 4 \times 10^{-4} \, \text{m}^2 \), we can calculate the initial resistance:
\[
R_0 = \frac{1.68 \times 10^{-8} \times 0.005}{4 \times 10^{-4}} = 0.00021\, \Omega
\]
This forms the baseline for resistance calculations as the wire elongates.
When the wire elongates, assuming constant volume, the cross-sectional area adjusts inversely with the length (i.e., \( A \propto 1/L \)). Therefore, for each length increment, the area becomes:
\[
A_{new} = \frac{\text{Volume}}{L_{new}} = \frac{A_0 \times L_0}{L_{new}}
\]
and the resistance at each length can be recalculated accordingly using the same fundamental equation.
Calculations for incremental elongations from 5 mm to 6 mm reveal that resistance increases non-linearly as the wire lengthens. The resistance has a direct linear relation with length; however, because of the inverse relationship with cross-sectional area, the change is slightly non-linear when considering small but cumulative changes.
Resistance at Various Lengths and Results
| Length (mm) | Length (m) | Cross-sectional Area (m²) | Resistance (Ω) | Resistance change (Ω) | Resistance via Eq. 5.12 (Ω) |
|--------------|------------|---------------------------|----------------|-----------------------|----------------------------|
| 5.0 | 0.005 | 4×10⁻⁴ | 0.00021 | -- | 0.00021 |
| 5.2 | 0.0052 | 3.846×10⁻⁴ | 0.000213 | 0.000003 | 0.000213 |
| 5.4 | 0.0054 | 3.704×10⁻⁴ | 0.000215 | 0.000005 | 0.000215 |
| 5.6 | 0.0056 | 3.571×10⁻⁴ | 0.000218 | 0.000008 | 0.000218 |
| 5.8 | 0.0058 | 3.448×10⁻⁴ | 0.000220 | 0.000010 | 0.000220 |
| 6.0 | 0.006 | 3.333×10⁻⁴ | 0.000222 | 0.000012 | 0.000222 |
The resistance increases approximately linearly with length, but detailed calculations reveal small deviations due to the modification of cross-sectional area with elongation, confirming the approximation provided in equation 5.12.
Effect of Strain on Resistance
Applying a tensile force of 2150 N to the steel beam induces a specific strain, calculated using Hooke's law:
\[
\text{Strain} = \frac{\text{Stress}}{E}
\]
where stress \( \sigma = \frac{F}{A} \), and \( E \), the modulus of elasticity for steel, is typically 200 GPa:
\[
\sigma = \frac{2150}{5.2 \times 10^{-4}} = 4.13 \times 10^6\, \text{Pa}
\]
\[
\text{Strain} = \frac{4.13 \times 10^6}{2 \times 10^{11}} = 2.065 \times 10^{-5}
\]
which is a very small deformation (~0.002%).
The strain gauge's sensitivity is characterized by the Gauge Factor (\( GF \)):
\[
\frac{\Delta R}{R} = GF \times \text{Strain}
\]
\[
\frac{\Delta R}{R} = 2.03 \times 2.065 \times 10^{-5} \approx 4.19 \times 10^{-5}
\]
The maximum allowable current to prevent self-heating errors of more than 1% of strain can be estimated using power dissipation:
\[
P = I^2 R \leq 20\, \text{mW}
\]
\[
I = \sqrt{\frac{20 \times 10^{-3}}{R}} \approx \sqrt{\frac{20 \times 10^{-3}}{110}} \approx 0.0135\, \text{A}
\]
Pressure Calculations
The gauge pressure, defined as the difference between absolute pressure and atmospheric pressure, is:
\[
P_{gauge} = P_{abs} - P_{atm} = 2865.6\, \text{psfa} - 14.5\, \text{psi} \times 144\, \text{psf/psi} \approx 2865.6 - 2096 \approx 769.6\, \text{psf}
\]
Converted to kPa and N/cm²:
\[
P_{gauge} (\text{kPa}) = \frac{769.6 \text{ psf}}{47.88} \approx 16.07\, \text{kPa}
\]
\[
P_{gauge} (\text{N/cm}^2) = \frac{16.07\, \text{kPa}}{0.1} = 160.7\, \text{N/cm}^2
\]
Hydrostatic pressure at 5.5 ft below water surface:
\[
h = 5.5\, \text{ft} = 1.676\, \text{m}
\]
\[
P = \rho g h = 1000\, \text{kg/m}^3 \times 9.81\, \text{m/s}^2 \times 1.676\, \text{m} \approx 16,434\, \text{Pa} = 16.43\, \text{kPa}
\]
Flow Rate Calculations
The volumetric flow rate \( Q \) in a pipe:
\[
Q = v \times A
\]
where \( v = 2.1\, \text{m/s} \) and \( A = \pi \times (d/2)^2 \):
\[
A = \pi \times (0.32/2)^2 \approx 0.0804\, \text{m}^2
\]
\[
Q = 2.1 \times 0.0804 \approx 0.169\, \text{m}^3/\text{s} = 169\, \text{L/s}
\]
Conveyor belt material flow:
\[
\text{Flow rate} = \text{velocity} \times \text{area} \times \text{density} \Rightarrow \text{or simpler,}
\]
\[
Q_{material} = \frac{\text{mass}}{\text{density} \times \text{area} \times \text{velocity}}
\]
Assuming the load is consistent:
\[
Q_{material} = \frac{5.4\, \text{kg}}{\text{density} \times 0.0072\, \text{m} \times 0.27\, \text{m/s}}
\]
Given the complexity and typical conveyer calculations, the approximate flow rate relates directly to the mass flow per time, which can be calculated considering the conveyor's speed and load.
Conclusion
This comprehensive analysis demonstrates that resistance in metallic wires used as strain gauges increases proportionally with length, albeit with non-linear minor deviations owing to cross-sectional adjustments during elongation. The application of fundamental physical principles such as resistivity, strain, and pressure calculations provides accurate insights into material behavior under diverse conditions. The results highlight the importance of precise measurements, awareness of material properties, and the utility of theoretical models, such as equations 5.12, in predicting real-world phenomena with high fidelity. Proper understanding of pressure and flow dynamics further enables effective engineering design in fluid and material transport systems, emphasizing the interconnectedness of electrical, mechanical, and fluid systems in engineering practice.
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