A wire supports a weight of 3W, what should its diameter be to stretch only 0.100mm, as in the scenario below.
A metal wire of diameter D stretches by 0.100mm when supporting a weight W. If a wire of the same length is used to support a weight equal to 3W, what should its diameter have to be (in terms of D) so that it still stretches only 0.100mm?
The amount by which a wire stretches is directly proportional to the weight pulling the wire and the length of the wire. This amount is also inversely proportional to the cross sectional area of the wire. As the cross sectional area of a round wire is proportional to its diameter. This means that the amount of stretch is inversely proportional to square of the diameter.
Thus when the effect of only increasing the weight from W to 3W without changing the length or diameter of the wire the amount of stretch also will triple from 0.1 mm to 0.3 mm.
To reduce this stretch back to 0.1 mm by only changing the diameter without altering either the weight or the length of wire, the diameter should be increased so that cross sectional area increases three fold.
The area will increase 3 fold when:
(Increased diameter/D)^2 = 3
Increased diameter = D*3^(1/2)