When a weight `W` is hung from one end of a wire of length `L` (other end being fixed), the length of the wire increases by `l`. If the same wire is passed over a pulley and two weights `W` each are hung at the two ends,what will be the total elongation in the wire?

Explanation

In case a let `Y` be Young's modulus of the material of the wire. If a is its area of cross section, then we can write
`y=(F//aF//a)/(l//L l//L)=(WL)/(al)(:'F=W)`
Increase in length of wire `l=(WL)/(aY)`
In case b we can treat either segment of length `L//2` in fig. as if one to its ends is ficed while the other end is attached to the load `W` because the point `M` remains stationary i.e., fixed all the time due to symmetrical load. Thus, when the wire is passed over the pulley, let `l'` be the increase in the length of each segment. Since each segment is of length `L//2`, we have
`Y=(W(L//2))/(al')`
Increase in length of wire of one side of pulley `l'=1/2 (WL)/(aY)=l/2`
Therefore, increase in the length of both segments of the wire `=l'+l'=l/2+l/2=l`
so the increase in length remains the same.