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

Options

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

Answer

C

Explanation

If L is the original length of wire, then
change in length of first wire, `Delta L_(1) = (L_(1) -L)`
change in length of second wire, `Delta L_(2) = (L_(2) -L)`
Now, Young.s modulus, `Y= (T_(1))/(A) xx (L)/(Delta L_(1)) = (T_(2))/(A) xx (L)/(Delta L_(2))`
or `(T_(1))/(Delta L_(1)) = (T_(2))/(Delta L_(2)) or (T_(1))/(L_(1) - L) = (T_(2))/(L_(2) - L)`
or `T_(1) L_(2) - T_(1) L = T_(2)L_(1) - LT_(2) or L= (T_(2) L_(1) - T_(1) L_(2))/(T_(2) - T_(1))`