The length of a metal wire is `L_1`, when the tension is `T_1` and `l_2` when the tension is `T_2`. The unstretched length of the wire is

Options

A. `(L_1+L_2)/2`
B. `sqrt(L_1L_2)`
C. `(T_2L_1-T_1L_2)/(T_2-T_1)`
D. `(T_2L_1+T_1L_2)/(T_2-T_1)`

Answer

C

Explanation

Young modulus , Y= `(stress)/( strai n)`
or, strain `= (stress)/Y`
If the length of the wire is `L_0` when there is no tension in the string, then in the first case , stress `=T_1/a` and strain =`(L_1-L_0)/L_0`. [a=area of cross section=constant approximately 0]
So, `(L_1-L_0)/L_0=T_1/(aY)or,1/(aY)=1/T_1(L_1/L_0-1)`
Similarly in the second case , `1/(aY)=1/T_2(L_2/L_0-1)`
So, `1/T_0(L_1/L_0-1)=1/T_2(L_2/L_0-1)`
or,`1/L_0=(L_1/T_1-L_2/T_2)=1/T_1-1/T_2`
or,`1/L_0=(T_2L_1-T_1L_2)/(T_1T_2)=(T_2-T_1)/(T_1T_2)thereforeL_0=(T_2L_1-T_1L_2)/(T_2-T_1)`