Before solving the problem, it is important to understand two important concepts:

1. The stress-strain relationship of a material: This describes the relationship between the force applied to a material and the resulting deformation. It is represented by a graph called a stress-strain curve.

2. Hooke's Law: This states that within the elastic limit of a material, the stress is directly proportional to the strain.


To solve the problem, we need to use Hooke's Law and the stress-strain relationship of the material. Here are the steps to follow:

1. Use Hooke's Law to find the stress in the wire:
- Stress = Force/Area
- The area of a wire is given by the formula for the area of a circle, A = πr^2, where r is the radius of the wire. Since the diameter of the new wire is four times that of the original wire, its radius is twice that of the original wire.
- Hence, the new wire has an area of 4πr^2, which is 4 times that of the original wire.
- Therefore, the stress in the new wire is the same as that in the original wire, which is 1 kN/mm^2.

2. Use the stress-strain relationship of the material to find the strain in the wire:
- The stress-strain relationship of a material is represented by a graph called a stress-strain curve.
- Since the stress in the new wire is the same as that in the original wire, the strain in the new wire is the same as that in the original wire.
- Let ε be the strain in the wire. Then, from the stress-strain curve, we know that:
- Stress = E×ε, where E is the modulus of elasticity of the material.
- Hence, ε = Stress/E.
- Since the stress in the wire is 1 kN/mm^2 and the modulus of elasticity of the material is a constant value, we can use the same value for both wires.

3. Use the strain in the new wire to find its elongation:
- Let L be the length of the wire.
- The elongation of the wire is given by the formula ΔL = ε×L.
- Since the strain in the new wire is the same as that in the original wire, we can use the same value for both wires.
- Therefore, the elongation of the new wire is four times that of the original wire, which is 4×1 = 4 mm.


The wire of the same material and length but of four times that of diameter would be stretched by the same force for 4 mm.