Roller Coaster Forces

Taking positive down, the force equation for a passenger upright at the top of the loop is: mg-F = ma = m((v^2)/R where F is the force from the seat. If upside down, the F term is positive. In either case, F can be set to zero with the proper choice of v.

I am not sure if I understand the question.

For an object to be in in circular motion there must be a net force on the object towards the center of the circle. The equation is (mv^2)/r.

If if the object is a "rider" on a roller coaster, the forces on her would be the seat pushing on her, and gravity. We have to do a vector addition of those two forces to get the net force. But we can think it through.

At the top of the loop, the seat is pushing down and gravity is pulling down. To get the net force required for circular motion, the force of the seat will be less. Gravity is providing a part (or all) of the net force.

At the bottom of the loop, the seat is pushing up and gravity is pulling down. Now to get the net force required, the seat has to put more force to counteract the force of gravity pulling in the opposite direction.