Coaster Dips and Hills Physics

Physics of Coaster Dips and Hills
The above discussion and force analysis applies to the circular-like motion of
a roller coaster car in a clothoid loop. The second section along a roller
coaster track where circular motion is experienced is along the small dips
and hills. These sections of track are often found near the end of a roller
coaster ride and involve a series of small hills followed by a sharp drop.
Riders often feel heavy as they ascend the hill (along regions A and E in the
diagram below). Then near the crest of the hill (regions B and F), their
upward motion makes them feel as though they will fly out of the car; often
times, it is only the safety belt that prevents such a mishap. As the car
begins to descend the sharp drop, riders are momentarily in a state of free
fall (along regions C and G in the diagram below). And finally as they reach
the bottom of the sharp dip (regions D and H), there is a large upwards force
that slows their downward motion. The cycle is often repeated mercilessly,
churning the riders' stomachs and mixing the afternoon's cotton candy into
a slurry of ... . These small dips and hills combine the physics of circular
motion with the physics of projectiles in order to produce the ultimate thrill of
acceleration - rapidly changing magnitudes and directions of acceleration.
The diagram below shows the various directions of accelerations that riders
would experience along these hills and dips.
Force Analysis of Coaster Hills
At various locations along these hills and dips, riders are momentarily
traveling along a circular shaped arc. The arc is part of a circle - these
circles have been inscribed on the above diagram in blue. In each of these
regions there is an inward component of acceleration (as depicted by the
black arrows). This inward acceleration demands that there also be a force
directed towards the center of the circle. In region A, the centripetal force is
supplied by the track pushing normal to the track surface. Along region B,
the centripetal force is supplied by the force of gravity and possibly even the
safety mechanism/bar. At especially high speeds, a safety bar must supply
even extra downward force in order to pull the riders downward and supply
the remaining centripetal force required for circular motion. There are also
wheels on the car that are usually tucked under the track and pulled
downward by the track. Along region D, the centripetal force is once more
supplied by the normal force of the track pushing upwards upon the car.
The magnitude of the normal forces along these various regions is dependent
upon how sharply the track is curved along that region (the radius of the
circle) and the speed of the car. These two variables affect the acceleration
according to the equation
a = v / R
and in turn affect the net force. As suggested by the equation, a large speed
results in a large acceleration and thus increases the demand for a large net
force. And a large radius (gradually curved) results in a small acceleration
and thus lessens the demand for a large net force. The relationship between
speed, radius, acceleration, mass and net force can be used to determine the
magnitude of the seat force (i.e., normal force) upon a roller coaster rider at
various sections of the track. The sample problem below illustrates these
relationships. In the process of solving the problem, the same problem-
solving strategy enumerated above will be utilized.