A COASTER'S LOOP
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.
Other loop possibilities
If you use or find this page useful or have any comments, please contact the author so maybe he'll do more. Author: Tony Wayne
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