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Now let's look at the acceleration function for circular motion, which is the second derivative of the rotational function with respect to time:

F(t)ᵣₒₜ = Re^(iwt)

F'(t)ᵥₑₗ = iwRe^(iwt)

F''(t)ₐ = -w²Re^(iwt)

The units here are R/w² which is dist/sec² for acceleration.

We notice that the function is being multiplied by the -1 operator. Which means the acceleration vector is opposite the original rotational vector on the real axis. When we take the tail end of the acceleration vector and attach it to the front tip of the rotational vector; we notice that the acceleration points inward towards center as the rotational vector traces a circle. This then is the centripetal acceleration which acts on an object in circular motion, keeping it from traveling in a straight line tangent to the circle. Because of Newton's third law of reactionary forces, the centripetal acceleration causes an equal and opposite acceleration called the centrifugal acceleration which is a real force. This equal and opposite force acts against the center source, pulling it outward, and is caused by the object in circular motion trying to travel in a straight line. 

So the centripetal acceleration acts against the object in circular motion, pulling it inwards and keeping it from traveling straight. And the centrifugal acceleration acts against the center source, pulling it outwards as the object in motion trys to travel in a straight line.


Consider the following:

F''(t)ₐ = -w²Re^(iwt)

-w²R = (-wR)²/R = (-Vorb)²/R

-Vorb²/R = -GM/R²

When we set the centripetal acceleration for orbital motion, equal to the centripetal acceleration for gravity; we are saying that gravity provides exactly the centripetal acceleration needed for orbital motion(Vorb).

We are not saying that the outward centrifugal acceleration and the inward gravitational acceleration both act on an object in orbit. Because the outward centrifugal acceleration acts against the center source, pulling it outward as the object in orbit tries to travel straight; and the inward centripetal acceleration acts against the object in orbit, pulling it inwards, and keeping it from traveling straight.



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