Why Does a Teardrop-shaped Roller Coaster Loop Improve Safety and Performance?

AI Thread Summary
Teardrop-shaped loops on roller coasters enhance safety and performance by optimizing the centripetal acceleration experienced by riders. In this design, the shape allows for a smoother transition through the loop, reducing the risk of riders being ejected from their seats. The maximum speed at the bottom of the loop is 31.0 m/s, while the speed at the top is 12.0 m/s, with a centripetal acceleration of 2g. This design ensures that the forces acting on the riders are manageable, contributing to a safer ride experience. Understanding the physics behind these loops is crucial for roller coaster engineering.
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Homework Statement


A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop. The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure that the cars remain on the track. The biggest loop is 40.0 m high, with a maximum speed of 31.0 m/s (nearly 70 mph) at the bottom. Suppose the speed at the top is 12.0 m/s and the corresponding centripetal acceleration is 2g.

What is the radius of the arc of the teardrop at the top?


Homework Equations


Ac=mv^2/r
Fr= n+mg


The Attempt at a Solution


I think n + mg = mv^2/r, but I don't know what to do with the given value Ac=2g or how to go any further from here. Any help is greatly appreciated.
 
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nvm, I got it.
 

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