# Banked Turn

## A turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn.

For a vehicle moving in a circle - like a train or a car in a curve - the wheels on the vehicle produces a centripetal acceleration toward the center of the circle. The road or track experiences a centrifugal thrust that tries to move the road or track outwards.

The outwards thrust can be reduced by inclining the outside of the track. The inclined angle - or banked angle - is the angle at which a vehicle is inclined about its longitudinal axis with respect to its path.

The banked angle can be calculated in radians as

*Θ _{rad} = tan^{-1}(v^{2} / (r a_{g})) (1)*

*where *

*Θ _{rad} = banked angle (rad)*

*v = velocity (m/s) *

*r = radius of the curve (m)*

*a _{g} = acceleration of gravity (9.91 m/s^{2}) *

- or alternatively in degrees

*Θ _{degrees} = tan^{-1}(v^{2} / (r a_{g})) (360 / 2 π) (1b)*

### Example - A Train on a Railway Track in a Curve with Radius *1000 m* with Speed *70 km/h*

The required banked angle to avoid the centrifugal force can be calculated:

*Θ = tan ^{-1}(((70 km/h) (1000 m/km) / (3600 s/h))^{2} / ((1000 m) (9.81 m/s^{2}))) *

*= 0.0385 rad*

*= (0.0385 rad) (360 / 2 π)*

*= 2.2 ^{o}*

### Banked Turn Calculator

This calculator can be used to calculate the centripetal acceleration of the car and the banked angle.

### Road Banked Turn Diagram

The diagram below indicates velocity vs. curve radius and required banked turn to compensate centrifugal forces.