You have landed a job in Disney’s Imagineering Department. (Congratulations!) There, you are part of a team that is building a new roller coaster. Several of your coworkers have submitted proposals. Now you have to sit down to select one design that will be prototyped and studied further. Today you are studying the first section of one of those designs. The car is to begin virtually at rest atop a hill. (The cars are just barely pulled over this initial hill.) From there, the car descends until it enters a vertical loop. Rather than a circular loop, this design has a teardrop, or clothoid, design. This shape can be thought of as a large radius circular arc at the bottom and a small radius arc at the top. (That is, the top and bottom can be thought of as portions with two different circular paths.) In this design, the bottom radius of curvature is 40m and the top is 20m. The top of the loop is 60m above the ground. There are a few safety considerations for the roller coaster: The normal force should always be less than 5.5 times the weight. (Disney's lawyers don't want riders blacking out.) The normal force should always be away from the track. (If it is ever zero or toward the track, the safety mechanism will not be able to hold the car on the track.) What is the range of possible hill heights that allow the ride to operate within the safety limits?
You have landed a job in Disney’s Imagineering Department. (Congratulations!) There, you are part of a team that is building a new roller coaster. Several of your coworkers have submitted proposals. Now you have to sit down to select one design that will be prototyped and studied further. Today you are studying the first section of one of those designs. The car is to begin virtually at rest atop a hill. (The cars are just barely pulled over this initial hill.) From there, the car descends until it enters a vertical loop.
Rather than a circular loop, this design has a teardrop, or clothoid, design. This shape can be thought of as a large radius circular arc at the bottom and a small radius arc at the top. (That is, the top and bottom can be thought of as portions with two different circular paths.) In this design, the bottom radius of curvature is 40m and the top is 20m. The top of the loop is 60m above the ground.
There are a few safety considerations for the roller coaster:
-
The normal force should always be less than 5.5 times the weight. (Disney's lawyers don't want riders blacking out.)
-
The normal force should always be away from the track. (If it is ever zero or toward the track, the safety mechanism will not be able to hold the car on the track.)
What is the range of possible hill heights that allow the ride to operate within the safety limits?
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