One way to look at is like this:
(mg) e_\rho+(N) e_\rho=-(m \omega^2 )e_r
where rho is the radial direction in spherical coordinates, and r is the radial direciton in cylindrical coordinates.
This equation can never be true since the LHS and RHS are in different directions no matter the value of N, so the assumption that the acceleration is purely centripetal is incorrect. But what other acceleration can there be? Assuming that the particle stays on the sphere, the equation must be:(mg) e_\rho+(N) e_\rho=-(m \omega^2) e_r+(x)e_\theta+(y)e_\phi
but there is no way to make this equation work unless y=0, since nothing points in the azimuthal/longitudinal direction (you can dot both sides with e_\phi to get this result), so only the polar (latitude) direction is left. So the equation is:
(mg) e_\rho+(N) e_\rho=-(m \omega^2) e_r+(x)e_\theta
You can solve for x, since there is only one value of x such that the RHS adds to something in the e_rho direction. Once you solve for x, you can solve N.
Anyways, this exercise has been interesting. I originally thought that friction had to be the key, so that you could have the equation:
(mg) e_\rho+(N) e_\rho+(F) e_\theta=-(m \omega^2 )e_r
In fact, the value of F would just be -x: i.e., you can either have friction to stop the movement, or have the movement.
However, if friction is really the reason we don't slide into the equator, then it should be true that ocean and air currents point towards the equator, since fluids are flowy and aren't affected by friction as much. But looking at air/water current maps, I didn't see such a flow towards the equator. I then reasoned that the equator is hotter since the sun shines perpendicularly on the equator, and hotter objects have more pressure so this would counter the current towards the equator: hence the reason there is no general trend of currents towards the equator.
Anyways, poor reasoning, as it turns out the answer is that the vector in front of the (N) is not e_rho. In fact, D_H already mentioned this on page 3 of this thread:
D H said:
From the perspective of a non-rotating observer moving alongside the Earth, a person standing still on the surface of the Earth is undergoing uniform circular motion. A net force is needed to maintain that circular motion. This net force is normal to and directed towards the Earth's rotation axis. The forces acting on this person are gravitation, directed downward, and the normal force, directed upwards. Due to the Earth's non-spherical shape the angle between these forces is not quite 180 degree. The net sum of these two forces is exactly equal to the net force needed to make the person keep following that uniform circular motion.
But it's nice to know that on a perfect sphere, everything flows to the equator, but then stops when the bulge is sufficient to unalign the normal force and gravity. This is the reason planets bulge at the equator and why there is no friction on us. So physics works.