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I always thought that the Earth bulges due to inertia, not forces.
Particles on the equator naturally want to go on a tangent. I tell my
students that every force must have a source!
When you go on the "Gravitron" (rotor) ride at the amusement park, you do
feel an outward force. It's simply the reaction force (you on wall) to the
centripetal force (wall on you). It's called a centrifugal force but it
acts on the wall, not you. Thus, it's not part of the free-body diagram
(on you).
"It makes the earth bulge."
That is barely hardly somwhat slightly sort of true.
Making such a
statement gives students a highly exaggerated picture of the truth.