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 $k\left(\dfrac{x}{L}\right)^n$, where $n$ 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 $n$?
Answer: (A) see figure
$$X_{CM} = \frac{\int_0^L x\,\lambda\,dx}{\int_0^L \lambda\,dx} = \frac{\int_0^L x^{n+1}dx}{\int_0^L x^n dx} = \frac{n + 1}{n + 2}L$$
At $n = 0$ (uniform rod), $X_{CM} = \dfrac{L}{2}$. As $n$ increases, the mass crowds towards $x = L$ and $X_{CM}$ rises steadily, approaching $L$ but never reaching it. The curve rises quickly at first and then flattens.
This is graph (1).
Solution by Sreeraj P, M.Sc Physics