Minimum Velocity in Friction Banked Turn

Description

A banked turn (aka. 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.

When considering the effects of friction on the system, once again we need to note which way the friction force is pointing. When calculating a maximum velocity for our automobile, friction will point down the incline and towards the center of the circle. Therefore, we must add the horizontal component of friction to that of the normal force. The sum of these two forces is our new net force in the direction of the center of the turn (the centripetal force).

Solving for minumum velocity we have the formula shown. Which provides the minimum velocity for the automobile with the given angle of incline, coefficient of static friction and radius of curvature.

Related formulas

Variables

vvehicle velocity (m/s)
rradius of curvature (m)
gStandard gravity
θangle of incline (degree)
μscoefficient of static friction (dimensionless) (dimensionless)