Center of Mass — Question 8

Question 8

A rod on $[0,1]$ has density $\rho_n(x)=(n+1)x^n$, for integers $n\ge0$. Find $\bar x_n$ and its limiting location as $n\to\infty$.

Question 8 – Solution

Step 1: Find the total mass. For $n\ge0$, $$\begin{align*} m_n&=\int_0^1\rho_n(x)\,dx =\int_0^1(n+1)x^n\,dx\\ &=(n+1)\left[\frac{x^{n+1}}{n+1}\right]_0^1=1. \end{align*}$$ Thus the density is normalized to have total mass $1$.

Step 2: Find the first moment about the origin. $$\begin{align*} M_n&=\int_0^1x\rho_n(x)\,dx =(n+1)\int_0^1x^{n+1}\,dx\\ &=(n+1)\left[\frac{x^{n+2}}{n+2}\right]_0^1 =\frac{n+1}{n+2}. \end{align*}$$

Step 3: Compute the center of mass. $$\bar x_n=\frac{M_n}{m_n}=\frac{n+1}{n+2} =1-\frac{1}{n+2}.$$

Step 4: Take the limit. $$\lim_{n\to\infty}\bar x_n =1-\lim_{n\to\infty}\frac{1}{n+2}=1.$$ The density becomes increasingly concentrated near the right endpoint. $$\boxed{\bar x_n=\frac{n+1}{n+2},\qquad \bar x_n\to1}$$