Deriving Max. Velocity for Banked Curve with Friction (Centripetal Acceleration)

[Novii]

Homework Statement

A circular curve of radius R in a new highway is designed so that a car traveling at speed v0 can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmin to vmax.

Derive formula for vmax, as a function of µ (coefficient of static friction), v0, and R.

Homework Equations

F=ma
centripetal acceleration = v²/R
Force of static friction = µs * N

The Attempt at a Solution

I tried expressing the angle of the bank with v0 and R first, then substituting it in for equations found for vmax:

θ=arctan((v0^2)/(9.8R))
x: sinθN + cosθFfr= ma= m (vmax^2)/R
y: mg + sinθFfr= cosθN = 9.8m + sinθ * µ* N

Where θ is the angle of elevation, N is normal force, and Ffr is friction force.

But the x and y equations turned out to be pretty complicated, and I'm not sure how to proceed now. Are my equations right?

 

[vela]

Your equations look fine. Rearranging them slightly, you get

$$\begin{align*} N\sin\theta + \mu N \cos \theta &= mv_{max}^2/R \\ N\cos\theta - \mu N \sin \theta &= mg \end{align*}$$

Try dividing the first equation by the second and go from there.