What Minimum Speed Keeps Passengers Safe in a 12m Radius Roller Coaster Loop?

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SUMMARY

The minimum speed required for passengers to remain safely in their seats at the top of a 12-meter radius roller coaster loop is determined by the balance of forces acting on them. At the top of the loop, the normal force (Fn) must equal the gravitational force (Fg) for passengers not to fall out, leading to the equation Fc = Fg - Fn. The centripetal force (Fc) is calculated using the formula Fc = mv²/r, where m is the mass of the passenger, v is the velocity, and r is the radius of the loop. Thus, the minimum speed can be derived from the relationship between these forces.

PREREQUISITES
  • Understanding of centripetal force (Fc) and gravitational force (Fg)
  • Familiarity with Newton's second law of motion
  • Basic knowledge of circular motion dynamics
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Calculate the minimum speed using the formula v = √(g * r) where g is the acceleration due to gravity.
  • Explore the effects of varying the radius on the minimum speed required for safety.
  • Investigate the role of friction and its impact on roller coaster design.
  • Learn about the engineering principles behind roller coaster safety mechanisms.
USEFUL FOR

Physics students, mechanical engineers, amusement park designers, and anyone interested in the dynamics of roller coasters and safety measures in amusement rides.

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Homework Statement



A roller coaster loop has a radius of 12 m. If the passengers are not to fall out at the top of the loop, what is the minium speed the car must have at the top.



Homework Equations



Fc = Fg-Fn
Fc= mv^2/r
Fg = mg

The Attempt at a Solution



I figure that for the passengers not to fall, Fn should be equal to Fg, but that gives Fc = 0 which doesn't make sense solving for v. What condition is necessary for the passengers not to fall out?

Thanks,
 
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What do you mean by Fn?
 
Lets rather consider just one passenger. The forces he/she would experience under such conditions are his/her weight, Fg, and the force normal to the tangential of the tract, Fn (the seat that the passsenger is sitting on pushes him/her).

The normal force points towards the inner side of the curvature of the track. It has to provide the force that makes the passenger go "around the bend" of the track (make him/her change direction towards the inner side of the curvature).

This force needs to be large if the curvature of the track is strong (small radius) and small if the curvature is weak.

The normal force will also be small if the speed of the passenger is small - it need not provide much acceleration towards the inside of the curvature if the speed of the passenger is small.
 
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