Q 11-08-012NEETJEE MainHard
A uniform wire of length $L$, density $\rho$ and Young's modulus $Y$ hangs vertically from a rigid support. The extension of the wire due to its own weight is
Answer: (D) $\dfrac{\rho g L^2}{2Y}$
A small piece $dx$ at distance $x$ from the bottom holds up the wire below it, which weighs $\rho A x g$. Its stretch is $\dfrac{\rho g x\,dx}{Y}$.
$$\Delta L = \int_0^L \frac{\rho g x}{Y}dx = \frac{\rho g L^2}{2Y}$$
This is the same as if the whole weight acted at the midpoint.
Solution by Sreeraj P, M.Sc Physics