Apparent weight = Normal force from the surface on the body.

At the bottom of a loop/valley: N = mg + $mv^2$/r (heavier, pressed into seat) At the top of a loop/hill: N = $mv^2$/r - mg (lighter, lifted from seat) Weightless condition: N = 0 when v = sqrt(gr) at the top

Roller coaster at top: If v > sqrt(gr), passengers feel pushed outward; if v < sqrt(gr), they feel pushed inward (need restraints).

Car on hill crest: At v = sqrt(gr), the car lifts off the road. Maximum safe speed on a hill of radius r is sqrt(gr).

Car in valley: Normal force is always > mg. No risk of losing contact, but passengers feel heavy.