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

Explanation

Let 2l be the original length of wire AB, i.e., L = 2l, when wire is pulled into shape ACB, the increase in length,
`DeltaL=(AC+CB)-AB`
`=2(l^(2)+d^(2))^(1//2)-2l`
longitudinal strain `=(DeltaL)/(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)/(pir^2)`
`therefore` Young's modulus,
`Y=("longitudinal stress")/("longitudinal strain")`
`(F//pir^2)/(d^2//2l^2)`
`therefore` Tension in the wire
`F=Yxxpir^(2)xx(d^2)/(2l^2)`