It may be useful to recall that kinematics from a wide class of
"accelerated frames" is usefully analyzed into two different effects:
1) effects that arise from referring kinematical quantities to an
ACCELERATING ORIGIN, and
2) effects that arise from referring kinematical quantities to a ROTATING
COORDINATE SYSTEM anchored to the above accelerating origin.
I will refer to the combined effects 1) and 2) as effects referred to a
"rotating/accelerating frame".
The acceleration of a particle as referred to a FIXED coordinate system,
which is anchored to an INERTIAL origin, will differ from the acceleration
of that particle as referred to the above rotating/accelerating frame by
four terms:
a) the effect of 1) is simply to add Ao (the acceleration of the origin)
b) the effect of 2) is to add a Coriolis acceleration, a "transverse"
acceleration, and a centripetal acceleration.
These 4 "inertial" accelerations must be added to the acceleration as
referred to the rotating/accelerating frame in order to calculate the
acceleration as referred to a fixed coordinate system anchored to an
inertial frame. Quantitatively:
R'' = r'' + Ao + 2w x r' + w' x r + w x (w x r)
The notation should be apparent - consult Fowles' text for details (or ask).