What forces are involved with Earth's rotational bulge?

[AntiElephant]

I'm trying to understand mathematically, if possible, why it is that the Earth bulges at the equator as a result of its rotation and how exactly gravity manages to keep it all together. Would the better approach be to keep myself in a rotating frame of reference? I lack some knowledge of Netwon's Laws in non-inertial frames of reference but maybe just enough to understand what is going on here.

I want to focus on a point (a "piece" of Earth's matter) on the Earth's surface, along the equator, to understand why it bulges outwards. If the Earth was initially stationary and spherical, then the only force acting on this piece would be gravity $F_{grav}$. As the Earth gradually begins to rotate a centrifugal force $F_{centrif}$ appears pointing in a direction outwards, opposite to the axis of rotation, a fictitious force as a result of being in a non-inertial frame of reference.

The total force on this piece would be $F_{grav} - F_{centrif}$. If $F_{centrif} <= F_{grav}$ then surely the piece would still have a resulting force pointing towards the centre of the Earth and the Earth would remain spherical? If/once $F_{centrif} > F_{grav}$ then the piece would "fly" off from the Earth (in a stationary frame, this would be a result of inertia). Have I looked at this too simplistically? How, then is it that the Earth bulges and instead isn't at either of the extremes - either spherical or stuff "flying" off as a result of inertia?

 

[Bandersnatch]

You can't do this without considering directions of force vectors.
Draw a free body diagram for a point on the equator, another on one of the poles, and one in between.
The key is that the centrifugal and gravitational forces each point somewhere else.

 

[russ_watters]

I think the key that is being missed here is this:

If the Earth was initially stationary and spherical, then the only force acting on this piece would be gravity $F_{grav}$.

If a mass has only one force acting on it, it accelerates. So if the Earth is stable/stationary and gravity is pulling a mass down, what is pushing it up to keep if from falling to the center of the earth? What's the other force you are missing?

 

[stedwards]

You know, this is hardly a trivial problem, and especially so, if we did not initially know that the gravitating body was an oblate spheroid. Deriving the gravitational potential requires a fairly involved integral. The direction of the gravitational acceleration is not directly away from the center of mass.

Last, if we want the solution to include a variation in density with depth, we need to establish equipotential surfaces in the interior of the spheroid.

 

[tfr000]

Simplest explanation:
The poles of a spinning sphere are stationary except for rotation. The equator is the most rapidly rotating. The rapid rotation partially offsets gravity via centrifugal force. This tapers from equator to pole.

 

[A.T.]

The poles of a spinning sphere are stationary except for rotation. The equator is the most rapidly rotating.

The rotation (angular velocity) is the same for both. The radius and linear velocity are different.