A wire of radius r stretched without tension along a straight line is tightly fixed at A and B. fig. What is the tension in the wire when it is pulled in the shape ACB? Assume Uoung's modulus of material of the wire to by Y.

Explanation

Let 2 l be the original length of wire AB, i.e., `L=2l`. When wire is pulled into shape ACB, the increases in length.
`Delta L= (AC+CB)-AB=2 (l^(2)+d^(2))^(1//2) - 2l`
Londitudinal strain = `(Delta L)/(L)`
`=(2(l^(2)+d^(2))^(1/2)-2l)/(2l)`
`=(2l[1+d^(2)//l^(2))^(1/2)-1)/(2l)`
`=[1+1/2 (d^(2))/(l^2)-1] = (d^2)/(2l^2)`
Longitudinal stress = (tension)/(area)` = F/(pi r^(2))`
`:. ` Young's modudus,
`Y=(l o n g i t u d i n a l" stress")/("longitudinal strain") = (F//pi r^(2))/(d^(2)//2l^(2))`
`:.` tension in the wire , `F=Y xx pi r^(2) xx d^(2)//2l^(2)`.