Q: The length of a steel wire is *l*_{1} when the stretching force is T_{1} and *l*_{2} when the stretching force is T_{2}. The natural length of the wire is

(a) $ \displaystyle \frac{l_1 T_1 + l_2 T_2 }{T_1 + T_2}$

(b) $ \displaystyle \frac{l_2 T_1 + l_1 T_2 }{T_1 + T_2}$

(c) $ \displaystyle \frac{l_2 T_1 – l_1 T_2 }{T_1 – T_2}$

(d) $ \displaystyle \frac{l_1 T_1 – l_2 T_2 }{T_1 – T_2}$

Ans: (c)

Sol:

$ \displaystyle T_1 \propto (l_1 – l_0 )$ …(i)

$ \displaystyle T_2 \propto (l_2 – l_0 )$ …(ii)

On dividing

$ \displaystyle \frac{T_1}{T_2} = \frac{(l_1 – l_0 )}{(l_2 – l_0 )} $

$ \displaystyle T_1 (l_2 – l_0) = T_2 (l_1 – l_0) $

$ \displaystyle T_1 l_2 – T_1 l_0 = T_2 l_1 – T_2 l_0 $

$ \displaystyle T_1 l_2 – T_2 l_1 = l_0 (T_1 – T_2) $

$ \displaystyle l_0 = \frac{T_1 l_2 – T_2 l_1}{T_1 – T_2} $