The length of a metal wire is `l_1` when the tensionin it is `T_1 and is l_2` when the tension is `T_2`. The natural length of the wire is

Options

A. `(L_1 + L_2)/(2) `
B. `sqrt(L_1 L_2)`
C. `(T_2L_1 - T_1L_2)/(T_2 - T_1)`
D. `(T_2L_1 -T_1 L_2)/(T_2 - T_1)`

Answer

C

Explanation

Let the unstretched length of the wire be L. Area of cross-section of the wire = A (say)
Young.s modulus, Y = Stress / Strain
Case I: Length of wire = `L_1`, Tension in the wire = `T_1` Extension in the wire =`(L_1 - L)`
` therefore Y = (T_1//A)/((L_1 - L)//L) = (T_1L)/(A(L_1 - L)) ` ...(i)
Case II : Length of wire ` = L_2`
Tension in wire ` = T_2` , extension in the wire ` = (L_2 - L)`
` therefore Y = (T_2//A)/((L_2 - L)//L) = (T_2L)/(A(L_2 - L))` ..(ii)
From eqn (i) and (ii) `(T_1L)/(A(L_1 - L)) = (T_2L)/(A(L_2 - L))`
`T_1 (L_2 - L) = T_2 (L_1 - L)` or `T_1 L_2 = T_1 L = T_2 L_1 - LT_2`
`L(T_2 - T_1) = T_2 L_1 - T_1 L_2`
` therefore L = (T_2L_1 - T_1L_2)/(T_2 - T_1)`