Question

A long thin rod of length L has a linear density y(x)= Ax where x is the distance from left end of the rod

1- how far is the center of mass of the rod from the left end of the rod?

2- what is the mass of the rod?

Answer 1

1.

$$x_c=\frac{\int_0^Lx\rho(x)dx}{\int_0^L\rho(x)dx}$$

$$x_c=\frac{A\int_0^Lx^2dx}{A\int_0^Lxdx}=\frac{L^3/3}{L^2/2}=\frac{2L}{3}$$

2.

$$m=\int_0^L\rho(x)dx=A\int_0^Lxdx=\frac{AL^2}{2}$$