Derive the terrestrial equation of motion in the body-fixed frame

[richyw]

Homework Statement

Assume that the center of mass of the Earth moves with approximately constant velocity with respect to the fixed stars, and that $\mathbf{\omega}$, the angular velocity of the earth, is constant. Rederive the terrestrial equations of motion
$$\mathbf{F}'+m\mathbf{g}$$$$m\ddot{\mathbf{r}}-2m\mathbf{\omega}\times\dot{\mathbf{r}}$$
By writing Newton's laws in a body-fixed frame with the origin at the surface of the earth.

Homework Equations

For a particle with coordinates $\mathbf{r}_0$ in the inertial frame and $\mathbf{r}$ in the body-fixed frame, where the origin of the body fixed frame is at the instantaneous point $\mathbf{a}$ with respect to the inertial frame.
$$\mathbf{r}_0 = \mathbf{a}+\mathbf{r}$$

The Attempt at a Solution

I'm pretty confused on where to start here. I know that in the inertial frame $\mathbf{F}=m\mathbf{A}$. Could anyone help get me started, or point me towards some literature on this?

 

[richyw]

wait I think I just arrived at the first equation.