Q. Two wires of length $l$, radius r and length $2l$, radius 2r respectively having some Young’s modulus are hung with a weight $mg$. Net elongation is
A) $\frac{3\ mgl}{{\pi r}^{2}Y}$
B) $\frac{2\ mgl}{3{\pi r}^{2}Y}$
C) $\frac{3\ mgl}{{2\pi r}^{2}Y}$
D) $\frac{3\ mgl}{{4\pi r}^{2}Y}$
Explanation
(c) Young’s modulus $Y = \frac{mgl}{a_{1}l_{1}}$
${\ l}_{1} = \frac{mgl}{Y\pi r^{2}}$ ……..(i)
and $Y = \frac{mg(2l)}{a_{2}l_{2}} = \frac{mg(2l)}{\pi(2\ r)^{2}l_{2}}$
Or $l_{2} = \frac{mg\ l}{2Y\ \ \pi r^{2}}$ .……..(ii)
From Eqs. (i) and (ii), we have
$\therefore\ l_{1} + l_{2} = \frac{mgl}{Y\ \ \pi r^{2}} + \frac{mgl}{2Y\pi r^{2}} = \frac{3}{2}\ \frac{mgl}{Y\ \ \pi r^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.