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
Concept. For a rod with linear density $\lambda(x)$, $X_{cm}=\dfrac{\int x\lambda\,dx}{\int\lambda\,dx}$.
Step 1. Integrate with $\lambda\propto x^n$.
$$X_{cm}=\frac{\int_0^Lx^{n+1}dx}{\int_0^Lx^ndx}=\frac{L^{n+2}/(n+2)}{L^{n+1}/(n+1)}=\frac{n+1}{n+2}L$$
Step 2. Shape. At $n=0$: $L/2$. As $n$ grows the mass crowds towards $x=L$: $X_{cm}$ rises and approaches $L$, flattening out.
Answer. Option A (rising from $L/2$ towards $L$).
Solution by Sreeraj P, M.Sc Physics
