Q. When the tension in a metal wire is $T_{1},\ \ its\ length\ is\ l_{1}$. When the tension is $T_{2}\ its\ length\ is\ l_{2}.$ The natural length of wire is
A) $\frac{T_{2}}{T_{1}}\left( l_{1} + l_{2} \right)$
B) $T_{1}l_{1} + \ T_{2}l_{2}$
C) $\frac{l_{1}T_{2} - l_{2}T_{1}}{T_{2} - T_{1}}$
D) $\frac{l_{1}T_{2} + l_{2}T_{1}}{T_{2} + T_{1}}$
Explanation
(c) $Y = \frac{Fl}{A\mathrm{\Delta}l}\ $
$Y,\ l\ and\ A$ are constants.
$\therefore\ \ \frac{F}{\mathrm{\Delta}l} = constant\ \ \ \ or\ \mathrm{\Delta}l\ \propto \ F\ $
Now, $l_{1} - \ l\ \propto T_{1}\ and\ \ l_{2} - l\ \propto T_{2}$
Dividing, $\frac{l_{1} - l}{l_{2} - l} = \frac{T_{1}}{T_{2}}$
Or $l_{2}T_{2} - \ lT_{2} = l_{2}T_{1}\ \ or\ l\left( T_{1} - T_{2} \right) = l_{2}T_{1} - \ l_{1}T_{2}$
Or $l = \ \ \frac{l_{2}T_{1} - l_{1}T_{2}}{T_{1} - T_{2}}\ \ or\ \ \ \ l = \ \frac{l_{1}T_{2} - l_{2}T_{1}}{T_{2} - T - 1\ }$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.