Derivation [1]
In the fixed coordinate system we know from
Newton's second law that
,
where 
is a vector from the origin to an object. We shall consider the
object to be a point mass to reduce the equation to its general form.
Here, 
is an external force or the sum of external forces acting on a body
(i.e. air friction, tension on a pendulum bob, etc.)
Substituting this into the previously derived expression for forces in a rotating coordinate system, we arrive at the following equation.
The force terms arrived at earlier:
(B1)
Combining expressions (B1)
and (B2).
(B3)
Solving for the effective force.
(B4)
Combining forces.
(B5)
Generalizing the gravitational acceleration for a
body of any radius or mass.
(B6)
Resulting in the complete expression of Newton's
second law for the surface of any solid body.
(B7)
References
[1] Mechanics: Third Edition, by Kieth
R. Symon, pages 280 & 281
(B2)