The length of a metal wire is `l_(1)` when the tension in it is `F_(1)` and `l_(2)` when the tension becomes `F_(2)`. Find the natural length of wire.

Explanation

Let `l` be the nautral length of the wire for the force `F_(1)` if elongation is `e_(1).e_(1) = (F_(1)l)/(AY) rArr Y = (F_(1)l)/(Ae_(1))`
Similarly for the force `F_(2)` if elongation is `e_(2)`
`e_(2) = (F_(2) l)/(AY) rArr Y = (F_(2)l)/(Ae_(2)), l_(1) = l + e_(1)` and `l_(2) = l + e_(2)`
`Y = (F_(1)l)/(A(l_(1) - l)) = (F_(2)l)/(A(l_(2) - l))`
`rArr F_(1) (l_(2) - l) = F_(2) (l_(1) - l)` and `F_(1)l_(2) - F_(1)l = F_(2) l_(1) - F_(2)l`
`(F_(2) - F_(1))l = l_(1) F_(2) - l_(2)F_(1)` or `l = (l_(1) F_(2) - l_(2) F_(1))/((F_(2) - F_(1)))`