Q. A thin rod of length $L$ is lying along the $x$-axis with its ends at $x\ = \ 0$ and $x\ = \ L$. Its linear density (mass\length) varies with $x$ as k$\left( \frac{x}{L} \right)^{n}$, when $n$ can be zero or any positive number. If the position $x_{CM}$ of the centre of mass of the rod is plotted against $n$, which of the following graphs best approximates the dependence of $x_{CM}$ on $n$?
A) [image:7_image63.emf]
B) [image:7_image64.emf]
C) [image:7_image65.emf]
D) [image:7_image66.emf]
Explanation
(a) $x_{CM} = \frac{\int_{}^{}{xdm}}{\int_{}^{}{dm}}$
[image:7_image162.emf]
If $n = 0$
Then $x_{CM} = \frac{L}{2}$
As $n$ increases, the centre of mass shift away from $x = \frac{L}{2}$ which only option (a) is satisfying.
Alternately, you can use basic concept.
$x_{CM} = \frac{\int_{0}^{L}{k\left( \frac{x}{L} \right)^{n}} \times xdx}{\int_{0}^{L}{k\left( \frac{x}{L} \right)^{n}dx}}$
$= L\left\lbrack \frac{n + 1}{n + 2} \right\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.