NSW · HSCModule 5
Banking angle calculator
Compute the design banking angle for a curve given speed and radius, or the friction-free design speed given a banked angle and radius.
Inputs
Result
Banking angle θ
34.23degrees
tan θ = v²/(rg)
0.6803
tan θ = v²/(rg). At the design speed no friction is needed.
How this calculator works
On a banked track with no friction, the normal force N is perpendicular to the road. Its vertical component balances gravity (N cos θ = mg), and its horizontal component provides the centripetal force (N sin θ = mv²/r). Dividing gives the design relationship tan θ = v²/(rg).
See the worked example in our non-uniform circular motion dot point answer.
Common questions
- What is the banking angle formula?
- tan θ = v² / (rg). At this angle, the horizontal component of the normal force alone provides the centripetal force, with no friction required.
- Why is mass not in the formula?
- Both the gravitational pull and the required centripetal force scale with the mass of the vehicle, so mass cancels. The design speed depends only on radius, gravity and angle.
- What happens if a car drives faster than the design speed?
- Friction must act down the slope to provide the extra inward force. If the speed exceeds friction's limit, the car slides outward.
- Does this work for aircraft turns?
- Yes. A banked turn in an aircraft uses the horizontal component of the lift force in the same way that a banked road uses the horizontal component of the normal force. Replace N with L and the formula is identical.