Final answer:
To maintain the same stretching distance as the length of the wire is doubled, the new wire must have a diameter that satisfies the constant volume condition. The new diameter should be √2 times the original, which gives approximately 1.4 mm.
Step-by-step explanation:
The problem presented is related to material strength and elasticity, specifically pertaining to Young's modulus, which relates stress and strain in a linear material.
The question seeks to find out the necessary diameter a wire needs to be in order to maintain the same stretching distance when its length is doubled, implying a question related to cross-sectional area and its impact on material properties.
If a wire is doubled in length without changing how far it stretches, we have to consider that the volume of the material remains constant because the amount of stretching indicates the same degree of force applied.
Since the volume (V) is constant and is given by V = πr2l (where r is the radius and l is the length), doubling the length (2l) would require us to find a new radius that maintains the original volume.
This leads us to solve for r in V = πr2(2l), which yields a new radius that is √2 times the original radius.
Given the diameter is twice the radius, we must also double this effect for the diameter, leading us to a diameter that is √2 times the original.
Starting with a 1.0 mm diameter wire, the new diameter must be 1.0 mm * √2 approximately equal to 1.4 mm.
Thus, the wire should have a diameter of 1.4 mm to maintain the extent of stretch when its length is doubled.