The wire stretches by approximately 1.94 × 10⁻² m while the objects are in motion.
Calculate the initial length of the wire:
The initial length of the wire is given as 1.80 m.
Calculate the cross-sectional area of the wire:
The cross-sectional area of a cylinder is given by the formula:
A = πr²
where r is the radius of the cylinder. In this case, the diameter is given as 4.00 mm, so the radius is 2.00 mm. Converting millimeters to meters, we have r = 2.00 × 10⁻³ m. Substituting into the formula for area, we get:
A = π(2.00 × 10⁻³ m)² ≈ 12.57 × 10⁻⁶ m²
Calculate the net force on the wire:
The net force on the wire is the difference between the weights of the two objects. Since the objects are moving in opposite directions, the net force is simply the sum of their weights. In this case, the net force is:
Fnet = (4.30 kg - 3.60 kg) × 9.81 m/s² ≈ 27.12 N
Calculate the stress on the wire:
Stress is defined as force per unit area. In this case, the stress on the wire is:
σ = Fnet / A ≈ 2.16 × 10⁷ Pa
Calculate the strain in the wire:
Strain is defined as the change in length divided by the original length. In this case, the strain in the wire is:
ε = σ / Young's modulus
where Young's modulus is the material property that describes the stiffness of the wire. For steel, Young's modulus is approximately 200 × 10⁹ Pa. Substituting into the formula for strain, we get:
ε ≈ (2.16 × 10⁷ Pa) / (200 × 10⁹ Pa) ≈ 1.08 × 10⁻²
Calculate the stretch in the wire:
The stretch in the wire is simply the product of the strain and the original length. In this case, the stretch is:
ΔL = ε × L ≈ (1.08 × 10⁻²) × (1.80 m) ≈ 1.94 × 10⁻² m .