The centripetal force is provided by two sources: the weight mg of the car, directed downwards, and the reaction FR of the rails. We have at point A
where a positive FR pushes the car down, a negative one pulls it up
Now the car rides on rails. At point A the rails are above the car and therefore it can only push up against them. The rails then, reacting to the force, must push it down, somewhat similar to the situation in "Objects at Rest", in section #18 on Newton's second law. Thus FR must be positive: if it were negative it would mean that the rails were pulling the car upwards, which they cannot do.
We thus require FR > 0, that is
or, after adding mg to both sides
This is the same result as was obtained using the centrifugal force: the problem can be solved in the outside frame of reference--but the process is a bit more complicated. The intuitive meaning is shown in the drawing.
- If all forces on the car ceased at point A, it would
continue along a straight line to point B,
in accordance with Newton's first law.
- If only gravity acted, it would follow a parabola
to point C.
- For the rails to exert a positive pressure,
they must constrain the car to a tighter curvature
than gravity alone, forcing it to move to point D.