| MadSci Network: Physics |
I love this question because I like the way you've thought through the consequences of your current understanding.
I'll cover your last questions first, because clarifying the vocabulary is important. A centripetal force is any force that is pulling an object in a circular or roughly circular path. Gravity can be the centripetal force on an object in orbit, and the elastic tension in muscle and string can supply the centripetal force on an object you're spinning around your head. Contrary to one of your statements, the centripetal force does not act in the opposite direction of gravity. If you are in a stable circular orbit, that means gravity is a centripetal force on you. The term only describes the behavior of a force, it is not a force in itself.
Now, centrifugal force is one of the "fictional" forces that arrises when you are in a rotating system, and is actually just the consequence of the principle that a body will travel in a straight line if no force is applied. We call this inertia. If you set a ball down on the seat of a car and then make a turn to the left, the ball will appear to roll to the right. Sitting in the car, it looks like something is "pushing" the ball to the right - a centrifugal force. Looking from outside the car, you would see the ball continuing to travel in a straight line with nothing pushing on it, while the car moves underneath it. The real forces are the friction between the tires and the road that causes the car to turn and the friction between the seat and your body that keeps you moving with the car.
The force of gravity is calculated on mass alone because you want to calculate the real force, that which is truly attributable to gravity. The same formula will apply to you on the surface, an astronaut in orbit, or a passing asteroid moving in some arbitrary direction not bound in our system.
When you instead want to quote the "surface gravity" of a planet, meaning the measured inward force on objects sitting on the surface, then you have to take into account that what you are measuring is in a sense the "net" force on you. What is left over beyond what is needed to keep you (and your scale) accelerating in the rotating frame? For the astronaut, the answer is g=0. Gravity balances the fictional centrifugal force. She's weightless, with no apparent acceleration in any direction. For someone on the equator, the answer is g=9.78. Gravity dominates. For someone standing on the asteroid as it swings into a curve around the earth, they feel a net centrifugal force pushing outward, away from the earth, just like you feel sitting in a turning car. g<0. The strength of that acceleration would be dependent on both the force of gravity and the speed of the asteroid. An independent, non-rotating observer would see only the force of gravity in every case, but would also be able to see *all* of the acceleration it causes, including the centripetal part that makes the other people travel in a circle.
In other words, the precise calculation of "surface gravity" or "g" has to include a fictional "centrifugal correction term" because in the rotating frame we don't see the real centripetal acceleration that is keeping us rotating.
At the equator, the centrifugal correction is about 0.34%. Pretty insignificant for most calculations. The response to the earlier question you reference, 855511961.Ph, is a bit subtle, because at first glance the earth's distortion should enhance rather than cancel the centrifugal effect. The increased distance from the center reduces the force of gravity and increases your velocity, increasing the centrifugal correction. If the respondent is correct, then this is canceled out and dominated by the small correction to the gravitational force calculation due to the distortion from a perfect sphere and the increased density on the equatorial plane. That makes sense, because otherwise the system would be unstable and the earth would spin itself apart... I think.
Anyway, I hope I at least helped you clarify your thiking about centripetal and centrifugal forces. Again, good question.
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