Banked curve

A banked turn (or banking turn) is a turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. For a road or railroad this is usually due to the roadbed having a transverse down-slope towards the inside of the curve. The bank angle is the angle at which the vehicle is inclined about its longitudinal axis with respect to the horizontal.

If the bank angle is zero, the surface is flat and the normal force is vertically upward. The only force keeping the vehicle turning on its path is friction, or traction. This must be large enough to provide the centripetal force, a relationship which can be expressed as an inequality, assuming the car is driving in a circle of radius r:

The expression on the right hand side is the centripetal acceleration multiplied by mass, the force required to turn the vehicle. The left hand side is the maximum frictional force, which equals the coefficient of friction μ multiplied by the normal force. Rearranging the maximum cornering speed is

Note that μ can be the coefficient for static or dynamic friction. In the latter case, where the vehicle is skidding around a bend, the friction is at its limit and the inequalities becomes equations. This also ignores effects such as downforce which can increase the normal force and cornering speed.

As opposed to a vehicle riding along a flat circle, inclined edges add an additional force that keeps the vehicle in its path and prevents a car from being "dragged into" or "pushed out of" the circle (or a railroad wheel from moving sideways so as to nearly rub on the wheel flange). This force is the horizontal component of the vehicle's normal force. In the absence of friction, the normal force is the only one acting on the vehicle in the direction of the center of the circle. Therefore, as per Newton's second law, we can set the horizontal component of the normal force equal to mass multiplied by centripetal acceleration:

Because there is no motion in the vertical direction, the sum of all vertical forces acting on the system must be zero. Therefore, we can set the vertical component of the vehicles's normal force equal to its weight:

Solving the above equation for the normal force and substituting this value into our previous equation, we get:

...
Wikipedia