Roller Coaster Physics

The Physics of Roller Coaster Loops
The most obvious section on a roller coaster where centripetal acceleration
occurs is within the so-called clothoid loops. Roller coaster loops assume a
tear-dropped shape that is geometrically referred to as a clothoid. A clothoid
is a section of a spiral in which the radius is constantly changing. Unlike a
circular loop in which the radius is a constant value, the radius at the bottom
of a clothoid loop is much larger than the radius at the top of the clothoid
loop. A mere inspection of a clothoid reveals that the amount of curvature at
the bottom of the loop is less than the amount of curvature at the top of the
loop. To simplify our analysis of
the physics of clothoid loops, we
will approximate a clothoid loop
as being a series of overlapping
or adjoining circular sections. The
radius of these circular sections
is decreasing as one approaches
the top of the loop. Furthermore,
we will limit our analysis to two
points on the clothoid loop - the
top of the loop and the bottom of
the loop. For this reason, our
analysis will focus on the two circles that can be matched to the curvature of
these two sections of the clothoid. The diagram at the right shows a clothoid
loop with two circles of different radius inscribed into the top and the bottom
of the loop. Note that the radius at the bottom of the loop is significantly
larger than the radius at the top of the loop.
As a roller coaster rider travels through a clothoid loop, she experiences an
acceleration due to both a change in speed and a change in direction. A
rightward moving rider gradually becomes an upward moving rider, then a
leftward moving rider, then a downward moving rider, before finally becoming
a rightward-moving rider once again. There is a continuous change in the
direction of the rider as she moves through the clothoid loop. And as learned
in Lesson 1 , a change in direction is one characteristic of an accelerating
object. In addition to changing directions, the rider also changes speed. As
the rider begins to ascend (climb upward) the loop, she begins to slow down.
As energy principles would suggest, an increase in height (and in turn an
increase in potential energy) results in a decrease in kinetic energy and
speed. And conversely, a decrease in height (and in turn a decrease in
potential energy) results in an increase in kinetic energy and speed. So the
rider experiences the greatest speeds at the bottom of the loop - both upon
entering and leaving the loop - and the lowest speeds at the top of the loop.
This change in speed as the rider moves through the loop is the second
aspect of the acceleration that a rider experiences. For a rider moving
through a circular loop with a constant speed, the acceleration can be
described as being centripetal or towards the center of the circle. In the case
of a rider moving through a noncircular loop at non-constant speed, the
acceleration of the rider has two components. There is a component that is
directed towards the center of the circle ( a ) and attributes itself to the
direction change; and there is a component that is directed tangent ( a ) to
the track (either in the opposite or in the same direction as the car's
direction of motion) and attributes itself to the car's change in speed. This
tangential component would be directed opposite the direction of the car's
motion as its speed decreases (on the ascent towards the top) and in the
same direction as the car's motion as its speed increases (on the descent
from the top). At the very top and the very bottom of the loop, the
acceleration is primarily directed towards the center of the circle. At the top,
this would be in the downward direction and at the bottom of the loop it
would be in the upward direction.