When the tension in a metal wire is `T_(1)`, its length is `I_(1)`. When the tension is `T_(2)`, its length is `I_(2)`. The natural length of wire is

Options

A. `(T_(2))/(T_(1)) (l_(1) + l_(2))`
B. `T_(1) l_(1) + T_(2) l_(2)`
C. `(l_(1)T_(2) - l_(2) T_(1))/(T_(2) - T_(1))`
D. `(l_(1) T_(2) + l_(2) T_(1))/(T_(2) + T_(1))`

Answer

c

Explanation

We have, `Y = (Fl)/(A Delta l)`, have and A are constants.
`:. (F)/(Delta l)` = constant or `Delta l prop F`
Now `l_(1) - l prop T_(1)` and `I_(2) - I prop T_(2)`
Dividing both, `(I_(1) -l)(l_(2) - l) = (T_(1))/(T_(2))`
Or `l_(1)T_(2) - lT_(2) = l_(2)T_(1) -lT_(1) implies l(T_(1) - T_(2)) = I_(2)T_(1) - l_(1)T_(2)`
or `l = (l_(2)T_(1) - l_(1)T_(2))/(T_(1) - T_(2)) = (l_(1) T_(2) - l_(2) T_(1))/(T_(2) - T_(1))`