Calculating Speed for Banked Racing Track

[superdave]

Homework Statement

Car is driven on a banked circular racing track. Radius R, angle theta.

a) For what speed is the track designed (if there were no friction the car would not slide)
b) If the track is wet, u=0.15, what are the minimum and maximum speeds car must be driven so it stays on the track.

Homework Equations

a=v^2/R (centripetal)
a=sin (theta) * g
a_f=u*g*cos(theta) (for part b, friction)

The Attempt at a Solution

Okay, here's what I don't understand.

Centripetal force pulls the car towards the center. And gravity pulls the car down, again towards the center. So though I know they oppose each other, I can't figure out why they would oppose each other since it seems they don't.

 

[Delphi51]

In my experience these banked road questions are best worked with from the point of view of the car, where you have a centrifugal force rather than a centripetal one. Further to make sense of it, you have to find the components of both forces that are normal to the road and parallel to it. The total normal force determines the friction force. Then you work with the 3 forces parallel to the road to find your answers.

 

[kerimek]

To answer part a), the track is designed for a specific speed at which the centripetal acceleration, given by the equation a=v^2/R, is equal to the gravitational acceleration, which is given by the equation a=sin(theta)*g. This means that the car is moving at a speed where the centripetal force and the force of gravity are balanced, allowing the car to travel along the banked track without sliding off.

For part b), we need to consider the effects of friction on the car. The minimum speed the car can travel at without sliding off the track is given by the equation a_f=u*g*cos(theta), where u is the coefficient of friction and g is the gravitational acceleration. This means that the car must be driven at a speed that produces a centripetal acceleration greater than or equal to the frictional force acting on the car. The maximum speed the car can travel at without sliding off the track is determined by the same equation, but with the maximum value of the coefficient of friction (u=0.15 in this case). This means that the car must be driven at a speed that produces a centripetal acceleration less than or equal to the maximum frictional force acting on the car.