Circular Motion and Radius of Curvature

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SUMMARY

The minimum speed required for a roller coaster to maintain passenger safety while upside down at the top of a circular path with a radius of curvature of 7.4 m can be determined using the centripetal force equation. The relevant equations are Ac = v²/R and ΣF = mv²/R, where Fg (gravitational force) must equal Fc (centripetal force) to prevent the car from falling off the track. By equating these forces and solving for velocity, the necessary speed can be calculated definitively.

PREREQUISITES
  • Understanding of centripetal force and gravitational force concepts
  • Familiarity with the equations of motion in circular dynamics
  • Ability to interpret free-body diagrams
  • Basic algebra skills for solving equations
NEXT STEPS
  • Study the derivation of centripetal acceleration equations
  • Learn about the implications of radius of curvature in circular motion
  • Explore examples of roller coaster physics and safety calculations
  • Investigate the effects of speed on passenger safety in roller coasters
USEFUL FOR

Physics students, engineering students, and anyone interested in understanding the dynamics of circular motion and safety in amusement park rides.

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
[tex]\SigmaF[/tex]=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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