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.