Max and min speed around banked curve

diablo2121

1. The problem statement, all variables and given/known data

A car enters a turn whose radius is R. The road is banked at angle Theta, and the friction coefficient is mu. Find the max and min speeds for the car to stay on the road without skidding sideways.

2. Relevant equations

W = mg
N = -W
Friction force = f = muN
Centripetal acceleration = ac = v^2/R

3. The attempt at a solution
Here's the force diagram I drew:


I have the following equations set up:
N = mgcosTheta f = mumgcosTheta W = mg
-W + N + f = 0
NcosTheta - fcosTheta = Fc = ac

So far, I think I've set up the problem correctly, but now I don't know where to go from here.

LowlyPion

Examine the cases separately. For instance at the velocity that it would slip up the curve, what must the V be greater than?

Likewise for the case where it would slip down the inclined curve what does the force drawing tell you about how to treat the terms?

diablo2121

If the car were to slip up the curve, then the friction force f, pointing inward, is overcome. Likewise, the car slipping inwards would imply that W is overcoming f, which should be pointing outwards. How do I relate these concepts into my equations?

LowlyPion

Just write it down. What is the normal force? What force opposes it for it to slip in one direction or the other. What do you do with the weight component that is the sinθ term? What must the velocity be greater or less than for each case?