How Does a Banked Curve Let a Car Turn Without Relying on Friction?
Banked Curves: How a tilted road supplies centripetal force at one design speed
One speed needs no friction. With the bank alone, the car keeps a level circle only if N cos θ = mg and N sin θ = mv²/R. Dividing, tan θ = v²/(gR): one speed, vdesign = √(gR tan θ). Mass cancels, so a truck and a motorbike share it.
Off that speed the bank cannot cope alone. Faster, mv²/R outgrows N sin θ, so the car slides up until the outer kerb pushes it back down the slope. Slower, it slides down onto the inner kerb, which pushes up the slope. Each kerb pushes along the road, as sideways tire friction does, which is why a real banked curve is safe over a range of speeds.
Centripetal force is not an extra force. Fc is the net inward force: the inward part of N, plus the level part of a kerb's push at the outer kerb, minus it at the inner. The tires only hold the speed. The 12 m road is small beside R, so every lane is taken at radius R.
Banked Curves & the Design Speed
A frictionless banked curve has exactly one "design speed" where a car rounds the turn without relying on friction at all. In this interactive, change the car's speed, the bank angle θ, and the radius R, and watch the road's normal force split into a vertical part (N cosθ) that balances gravity and a horizontal part (N sinθ) that acts as the centripetal force. The turn is stable only when the required force mv²/R exactly equals what the bank supplies, mg tanθ — giving v_design = √(gR·tanθ).
How to use it: set a bank angle and radius, then adjust the speed until the status reads "On Design Speed." Toggle Components to see N break into N sinθ and N cosθ, and use the force-vs-speed graph to spot where the required and supplied curves cross. Try going too fast or too slow to see the car drift up or down the bank.
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