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Banking of roads
Intermediate
Bank angle supplies centripetal force without friction at design speed.
tanθ = v²/rgOn a banked curve, the horizontal component of the normal force provides the centripetal force. With no friction, tanθ = v²/(rg); with friction, vmax = √(rg(μ + tanθ)/(1 − μ tanθ)).
For v = 20 m/s and r = 100 m: tanθ = 0.41, θ ≈ 22°.
Don't resolve forces along the slope when the acceleration is horizontal; use horizontal and vertical.
In more detail
A banked road has its surface tilted towards the centre of a curve, so the normal force provides part of the centripetal force. At the ideal speed, friction is not needed.
Real-world uses
- Racing tracks and motorway curves are banked to allow safe speeds.
- Velodromes use steep banking to keep cyclists on a circular path.
- Motorway bends are banked so that drivers at the design speed do not need friction to stay on the road.
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Frequently asked questions
What is the ideal banking speed?
Where tan θ equals v² divided by rg, with no friction needed.
Why does banking help at low speeds too?
Because the inward component of the normal force can still help, though friction must prevent sliding downhill.
Why do banked roads help at low speed?
At low speed the component of normal force is too large for the required centripetal force, so friction must push the car outwards to balance.
Related topics in Newton's Laws & Forces
Rocket propulsion · Conservation of momentum · Circular motion dynamics · Vehicle on flat circular turn · Conical pendulum · Vertical circular motion