# The length of a steel wire is l1 when the stretching force is T1 and l2 when the stretching force is T2

Q: The length of a steel wire is l1 when the stretching force is T1 and l2 when the stretching force is T2. 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

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

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

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

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

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