Question

The linear density of a thin rod is given by p =cx, where c is a constant and x is the distance

measured from one end. If the rod is of length b, find the center of mass.

Answer 1

The center of mass by definition

$$x_c=\frac{\int_0^b\rho(x)xdx}{\int_0^b\rho(x)dx}$$

$$x_c=\frac{\int_0^bx^2dx}{\int_0^bxdx}=\frac{b^3/3}{b^2/2}=\frac{2}{3}b$$