Circular Motion and Radius of Curvature

AI Thread Summary
To determine the minimum speed a roller coaster must travel when upside down at the top of a circle with a radius of curvature of 7.4 m, the centripetal force must equal the gravitational force acting on the passengers. A free-body diagram indicates that if gravitational force exceeds the required centripetal force, the passengers will fall out. The relevant equations include Ac = v^2/R and ΣF = mv^2/R. By setting the centripetal force equal to gravitational force, one can derive the necessary speed for safe passage. Understanding the radius of curvature is crucial for solving this problem effectively.
ThatMathGuy
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Homework Statement


Oh boy... Here we go...

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle so that the passengers will not fall out? Assume a radius of curvature of 7.4 m.

I honestly have nothing. To begin with, I drew a free-body diagram but got nowhere considering the teacher never explained 'radius of curvature'. I've tried searching on the Internet but all I get are a bunch of formulas in geodesy and aren't applicable to the problem. It's not in the book either (Giancoli PHYSICS Updated Edition 2009).


Homework Equations


Ac=v^2/R
\SigmaF=mv^2/R




The Attempt at a Solution


Tried setting up a free-body diagram and got nowhere.
 
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At the top of the circular motion, gravity must provide no more than the centripetal force needed to hold the car in circular motion. If Fg exceeds Fc, the car will fall off the track. So start with Fc = Fg, fill in the detailed formulas and solve for v.
 

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