On a ride that takes you through a big dip, at the bottom the difference between the normal reaction force and your weight acts as the centripetal force. So R - mg = m(v squared) / r

On a very long swing of length L is released from height h above the equilibrium point. Maximum speed is at the lowest point and can be worked out by: 0.5m(v squared) = mgh, therefore (v squared) = 2gh

The difference of forces at this point equals the centripetal force so R - mg = m(v squared) / L. But (v squared) = 2gh, therefore R - mg = 2mgh / L and when h = L the centripetal force equals twice the person's weight.

On a big wheel at the highest point the reaction of the seat and the weight of the person both act towards the center of the circle, so R + mg = m(v squared) / r, and therefore R = [m(v squared) / r] - mg. If v = gr, the R = 0 and there will be no force from the seat.

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