A COASTER'S LOOP
On a well designed roller coaster loop, the riders will not be able to sense when they are traveling upside down. This is done by making sure the force that is exerted on the rider is at least equal to the weight of the rider.

Centripetal force applied to the track depends on the velocity of the car and inversely to the radius. The formula is:

In order to apply enough centripetal acceleration the roller coaster car has to either be traveling very fast or the radius of the loop has to be made small. Most rides have a tall loop. A tall loop means a big radius. The problem is, as a car goes up, it slows down. The higher it goes, the slower it will be traveling over the top. In order to apply a centripetal force equal to gravity, 1 g, at the top, the car must be traveling extremely fast as the rider enters the loop. On some of the early round loops, the riders actually had their necks broken as a combination of the sudden rise in the loop as they entered at an extremely high rate of speed. As a compromise, the loops today are designed around an irregular shape called a klothoid or spiral of Archimedes. These irregular loops allow a circular figure whose radius changes.

For the advanced reader, the formula for the klothoid shape is:

The formula for the "Spiral of Archimedes" in polar form is

where "a" describes the magnitude of the spiral and the greek letter "theta" is the angle through which the spiral is formed. To make a loop, the spiral will have to be mirrored horizontally.

Nothing is perfect in engineering. These designs operate under ideal circumstances. In real life, the curves need to be tweaked into the right shape.

The Simple Irregular Loop
Sometimes it is not necessary to go into all the math to have a little fun with the irregular loop. These loops can be simulated using the combination of semi-circles of different radii.

Other loop possibilities