Banked Curve Design Reference Cheat Sheet

Key Facts

• Centripetal acceleration on a curve is $a_c = \frac{v^2}{r}$ and points toward the center of the circular path.
• The required centripetal force is $F_c = \frac{mv^2}{r}$, where $m$ is mass, $v$ is speed, and $r$ is curve radius.
• For a frictionless banked curve, the design condition is $\tan\theta = \frac{v^2}{rg}$.
• The frictionless design speed is $v = \sqrt{rg\tan\theta}$.
• The required bank angle for a target speed is $\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)$.
• For a flat unbanked curve, the maximum speed before slipping is $v_{\max} = \sqrt{\mu_s rg}$.
• On a banked curve with friction, static friction may act down the slope at high speeds and up the slope at low speeds.
• Mass cancels from ideal banked curve formulas, so the safe design speed does not depend on vehicle mass when all vehicles have the same tire friction conditions.

Vocabulary

Bank Angle

The angle $\theta$ between the road surface and the horizontal on a curved path.

Design Speed

The speed $v$ at which a vehicle can travel around a banked curve without needing friction to prevent slipping.

Centripetal Force

The net inward force $F_c = \frac{mv^2}{r}$ required to keep an object moving in a circular path.

Normal Force

The contact force perpendicular to a surface, often written as $F_N$.

Static Friction

The friction force that prevents slipping between surfaces and has a maximum value $f_s \leq \mu_s F_N$.

Radius of Curvature

The radius $r$ of the circular path followed by the vehicle through the turn.