Calculate the elongations of the cooper and aluminium wire . Diameters of Cu and Al wires are 1.4 mm and 1mm respectively

To calculate the elongations of copper (Cu) and aluminum (Al) wires, we need to apply the formula for elongation due to tensile stress, which is derived from Hooke's Law. The formula is given by:

Elongation Formula

The elongation (ΔL) of a wire can be expressed as:

ΔL = (F × L) / (A × Y)

• ΔL = elongation
• F = force applied
• L = original length of the wire
• A = cross-sectional area of the wire
• Y = Young's modulus of the material

Step-by-Step Calculation

Let’s break down the calculation into manageable steps:

1. Determine the Cross-Sectional Area

The cross-sectional area (A) of a wire can be calculated using the formula for the area of a circle:

A = π × (d/2)²

Where d is the diameter of the wire.

2. Calculate for Copper Wire

For the copper wire with a diameter of 1.4 mm:

• Convert diameter to meters: 1.4 mm = 0.0014 m
• Calculate radius: r = d/2 = 0.0014 m / 2 = 0.0007 m
• Cross-sectional area: A_Cu = π × (0.0007 m)² ≈ 1.538 × 10^-6 m²

3. Calculate for Aluminum Wire

For the aluminum wire with a diameter of 1 mm:

• Convert diameter to meters: 1 mm = 0.001 m
• Calculate radius: r = d/2 = 0.001 m / 2 = 0.0005 m
• Cross-sectional area: A_Al = π × (0.0005 m)² ≈ 7.854 × 10^-7 m²

4. Young's Modulus Values

Next, we need the Young's modulus (Y) values for both materials:

• Young's modulus for copper (Y_Cu) ≈ 110 GPa = 110 × 10⁹ Pa
• Young's modulus for aluminum (Y_Al) ≈ 70 GPa = 70 × 10⁹ Pa

5. Calculate Elongation

Assuming the same force (F) and original length (L) for both wires, we can express the elongation for each wire:

For copper:

ΔL_Cu = (F × L) / (A_Cu × Y_Cu)

For aluminum:

ΔL_Al = (F × L) / (A_Al × Y_Al)

6. Ratio of Elongations

Since both wires are subjected to the same force and length, we can find the ratio of elongations:

ΔL_Cu / ΔL_Al = (A_Al × Y_Cu) / (A_Cu × Y_Al)

Final Thoughts

By substituting the values we calculated for the areas and Young's moduli, you can find the specific elongations for both wires. This approach not only gives you the elongation values but also illustrates how different materials respond to stress based on their physical properties. If you have specific values for the force and length, you can plug those into the equations to get numerical results for the elongations of both wires.