Forces in Circular Motion showed that something must supply the inward force on a car rounding a curve. On a flat road only friction can. Tilting the road lets the normal force help, and Friction on Ramps gives the friction limits. You can drive one in the Circles and Gravity lab.
Why it works
The car moves in a horizontal circle, so its acceleration $v^2/r$ points horizontally toward the center. Choose horizontal and vertical axes, not axes along the slope. With no friction, the normal force (perpendicular to the road, tilted toward the center by the bank angle $\theta$) has parts: $$N\cos\theta = mg \;\;(\text{vertical: no acceleration}), \qquad N\sin\theta = \dfrac{mv^2}{r} \;\;(\text{horizontal}).$$ Dividing gives $\tan\theta = \dfrac{v^2}{rg}$.
Key idea
The design speed of a curve banked at $\theta$ is $v = \sqrt{rg\tan\theta}$: there no friction is needed. Faster, friction must point down the slope; slower, up the slope. With static friction up to $\mu_sN$: $$v_{\max}^2 = rg\,\dfrac{\sin\theta + \mu_s\cos\theta}{\cos\theta - \mu_s\sin\theta}, \qquad v_{\min}^2 = rg\,\dfrac{\sin\theta - \mu_s\cos\theta}{\cos\theta + \mu_s\sin\theta}.$$
Worked example: The design speed
A curve of radius $100\ \text{m}$ is banked at $15^\circ$. At what speed is no friction needed?
- $v = \sqrt{rg\tan\theta} = \sqrt{100(9.8)(0.2679)} = \sqrt{262.6}$.
- $v = 16.2\ \text{m/s}$ (about 58 km/h).
Answer: $16.2\ \text{m/s}$.
Worked example: The top speed with friction
For the same curve with $\mu_s = 0.40$, how fast can a car go without sliding up the bank?
- Friction points down the slope at $\mu_sN$. Vertical: $N\cos\theta - \mu_sN\sin\theta = mg$. Horizontal: $N\sin\theta + \mu_sN\cos\theta = mv^2/r$.
- Divide: $$v^2 = rg\dfrac{\sin 15^\circ + 0.4\cos 15^\circ}{\cos 15^\circ - 0.4\sin 15^\circ} = 980\cdot\dfrac{0.6452}{0.8624} = 733\ \text{m}^2/\text{s}^2$$
- $v_{\max} = 27.1\ \text{m/s}$.
Answer: $27.1\ \text{m/s}$.
Common mistake
Splitting the forces along and across the slope, as for a block on a ramp, and writing $N = mg\cos\theta$. On a ramp that works because the block accelerates along the slope. Here the acceleration is horizontal, so the forces across the slope don't balance; in fact $N = mg/\cos\theta$ at the design speed, larger than the weight. Check: your vertical forces must add to zero, because the car neither rises nor sinks.
Racetracks are steeply banked so cars can corner fast with little help from friction. On a road the bank is modest, because cars also crawl around the curve in traffic and must not slide down when the road is icy; that is the $v_{\min}$ condition.