Q. A non-uniform thin rod of length $L$ is placed along $X$-axis as such its one of ends is at the origin. The linear mass density of rod is $l = l_{0}x$. The distance of centre of mass of rod from the origin is
A) $\frac{L}{2}$
B) $\frac{2L}{3}$
C) $\frac{L}{4}$
D) $\frac{L}{5}$
Explanation
(b) The mass of considered element is
[image:7_image164.emf]
$dm = \lambda dx = \lambda_{0}\ xdx$
$\therefore\ \ x_{CM} = \frac{\int_{0}^{L}{x\ dm}}{\int_{}^{}{dm}} = \frac{\int_{0}^{L}{x(\lambda_{0}xdx)}}{\int_{0}^{L}{\lambda_{0}xdx}}$
$= \frac{\lambda\left\lbrack \frac{x^{3}}{3} \right\rbrack_{0}^{L}}{\lambda_{0}\left\lbrack \frac{x^{2}}{2} \right\rbrack_{0}^{L}} = \frac{\lambda_{0}\frac{L^{3}}{3}}{\lambda_{0}\frac{L^{2}}{2}} = \frac{2}{3}L$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.