A composite wire consists of a copper wire and a steel wire. The length of the copper wire is

Calculation of Load on a Composite Wire: A Step-by-Step Solution

A composite wire consists of a copper wire and a steel wire. The length of the copper wire is 2.2 m and the length of the steel wire is 1.6 m. When a load is applied, the composite wire stretches by 0.7 mm. The task is to calculate the load, given that the Young's modulus for copper is 1.1 × 10¹¹ Pa and for steel is 2.0 × 10¹¹ Pa.

Solution:

We will use the formula for the elongation of a wire under a load:

ΔL = (F * L) / (A * Y)

Where:

• ΔL is the elongation of the wire,
• F is the force (load) applied,
• L is the length of the wire,
• A is the cross-sectional area of the wire,
• Y is the Young's modulus of the material.

Given values:

• Diameter of wire = 3.0 mm = 3.0 × 10⁻³ m,
• Length of copper wire, L_Cu = 2.2 m,
• Length of steel wire, L_Steel = 1.6 m,
• Total elongation, ΔL_total = 0.7 mm = 0.7 × 10⁻³ m,
• Young's modulus for copper, Y_Cu = 1.1 × 10¹¹ Pa,
• Young's modulus for steel, Y_Steel = 2.0 × 10¹¹ Pa.

For each wire, the elongation is given by:

ΔL_Cu = (F * L_Cu) / (A * Y_Cu), ΔL_Steel = (F * L_Steel) / (A * Y_Steel)

The total elongation is the sum of the elongations of both wires:

ΔL_total = ΔL_Cu + ΔL_Steel

Substituting the values:

ΔL_total = (F * L_Cu) / (A * Y_Cu) + (F * L_Steel) / (A * Y_Steel)

Factor out the common terms:

ΔL_total = (F / A) * (L_Cu / Y_Cu + L_Steel / Y_Steel)

Now, solving for the force (F):

F = (ΔL_total * A) / (L_Cu / Y_Cu + L_Steel / Y_Steel)

Step-by-step Calculation:

1. Calculate the cross-sectional area:

A = π * (d / 2)² = π * (1.5 × 10⁻³)² = 7.07 × 10⁻⁶ m²

2. Substitute the values into the formula for force:

F = (0.7 × 10^-3) * (7.07 × 10^-6) / (2 × 10^-11 + 8 × 10^-12)

3. Simplifying the denominator:

Denominator = 2 × 10^-11 + 8 × 10^-12 = 2.8 × 10^-11

4. Final calculation:

F = (4.949 × 10^-9) / (2.8 × 10^-11) = 1.77 × 10² N

Therefore, the load (F) is approximately 177 N.