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I want to ask a question about strain in a steel wire that I read in a book titled Advanced Physics by John Miller.

A passage of the book on Materials reads:

Consider 2 steel wires being stretched by the same weight. One wire is double the length of the other.

They have the same cross-sectional area, and the forces are the same but the extensions are not - the longer wire stretches twice as much even though it has the same tensile stress because it is twice as long to start with.

This means it stretches by the same fraction of its original length, which we give the name "tensile strain."

I cannot seem to understand why the wire with the longer length stretches twice as much - I don't seem to see the connection between the wire's length and the extension it produces.

Can someone explain why?

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The stress applied to the wire is (in this theoretical example) the exact same.

There is the same force applied through the whole length of the wire (neglecting small effects due to loading mechanism and weight); and the wire in each case has the same diameter/area, so the stress is also the same in every part of both wires.

Because of the known stress-strain relationships, we know that in the linear-elastic range, stress is directly proportional to strain. Strain is a measure of relative elongation compared to a total length. So strain basically measures the ratio of elongation compared to original unstained length. For the simple strain considered here, we can define it as $$\epsilon = \frac {\Delta L}{L}$$

Because strain is a measure of ratio, and the stresses are the same, value of $\epsilon$ will be the same, therefore the change in length ($\Delta L$) will vary in proportion to the total length ($L$).

You can also consider this like springs in series. (A good question on that here) Basically, if you put several springs with the same stiffness together end to end, then apply a force to all of them (like by hanging a weight off them for example), the springs in series will have a lot more motion than a single spring of the same stiffness. This is because each element experiences the same load, and stretches the same amount. The more total elements you have, the more total stretch when identical loads are applied.

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You can understand this as follows: take the long wire and cut it in half. This will stretch like the short wire. Now connect the second part to the first. This one will also stretch by the same length. Together, the stretch amounts to twice the short wire.

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