Maximum Speed and Banked Turns in Circular Motion

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SUMMARY

The discussion focuses on the physics of a car navigating a banked circular turn, specifically addressing maximum speed and the effects of banking on friction. For Part A, it is established that there is indeed a maximum speed for the car to remain in contact with the road, which is influenced by the radius of the turn. In Part B, the car's mass is given as 1000 kg, with a turn radius of 50 m and a banking angle of 15 degrees. The coefficients of friction are specified as static (u_s = 0.15), kinetic (u_k = 0.08), and rolling (u_r = 0.02), leading to a determination of the speed range (v_min < v < v_max) necessary for the car to maintain its path on the wet road.

PREREQUISITES
  • Understanding of circular motion and centripetal acceleration (a = mv^2 / r)
  • Knowledge of forces acting on a body in motion on a banked curve
  • Familiarity with the concepts of static and kinetic friction
  • Ability to analyze free body diagrams in two and three dimensions
NEXT STEPS
  • Calculate the maximum speed for a car on a banked curve using the formula involving radius and banking angle
  • Explore the effects of different coefficients of friction on vehicle stability during turns
  • Learn about the dynamics of vehicles in circular motion, including the role of mass and radius
  • Investigate the implications of wet road conditions on vehicle handling and safety
USEFUL FOR

This discussion is beneficial for physics students, automotive engineers, and anyone interested in the dynamics of vehicles navigating circular paths, particularly in varying road conditions.

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Homework Statement



Part A) A car drives over a circular hill (hill radius is r). Is there a maximum speed that the car can have and still remain in contact with the road? If not, why not. If so, find an expression for the maximum speed.

Part B) Now, the car (with mass = 1000kg) enters a circular turn of radius, r=50m. The road is inwardly banked at an angle of 15 degrees. The road is wet so the coefficients of friction are: static u_s = 0.15, kinetic u_k = 0.08, rolling u_r = 0.02. Determine the range for the speed (v_min < v < v_max) that the car must maintain to stay on the circular turn.

Homework Equations



a = mv^2 / r
v = wr

The Attempt at a Solution



Very lost with this problem. I am assuming for part a that there is a maximum speed (this makes sense just thinking about it physically) but I don't know how to find this. I also can deduce that the greater the radius is, the greater the max speed will become. For part b, I tried drawing a free body diagram, but it seems to be in three dimensions because of the bank in the curve and I'm not sure how to account for that. Any help would be appreciated. Thanks.
 
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Please? I could really use some help.
 
Even if it's just A or B?
 

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