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A car takes a flat corner and relies entirely on friction to keep from sliding out. But race tracks, velodromes, and highway off-ramps tilt their corners inward. On a perfectly banked curve a car could round the bend on sheet ice — with zero friction. How can tilting the road replace friction?
A car takes a flat corner and relies entirely on friction to keep from sliding out. But race tracks, velodromes, and highway off-ramps tilt their corners inward. On a perfectly banked curve a car could round the bend on sheet ice — with zero friction. How can tilting the road replace friction?
On a flat curve, the only thing pulling a car into the turn is sideways friction; lose grip and you slide off. Banking the road tilts the normal force so that its horizontal component points toward the centre of the circle and supplies the centripetal force by itself. For one special 'design speed' the banking angle does the whole job and no friction is needed. Above or below that speed, friction makes up the difference — until even friction runs out.
On a banked curve the normal force is tilted so part of it points toward the circle's centre. At the frictionless 'design speed' the banking angle alone supplies the centripetal force, giving a clean relation between angle, speed, and radius.
Set up two force equations for a car on a frictionless banked road. Perpendicular to the centre (vertical): N cos θ = mg. Toward the centre (horizontal): . Dividing the second by the first cancels N and m: . Solving for speed gives the design speed v = √(rg tan θ). If the car goes faster than this, it tends to slide up and outward, so friction must act down the incline; if slower, it tends to slip down and inward, so friction acts up the incline. The banking angle is chosen for the most common speed so that wear and skidding are minimised. **Limiting case:** leaves friction to do everything (, the flat-road formula); at the design speed the normal force alone supplies the turn and friction is not needed at all. **Connect it:** nothing new is happening — it is the same as ordinary circular motion; banking simply tilts so a slice of it points at the centre.
Thinking a banked curve removes the need for friction at any speed — the bank fully replaces friction only at one design speed; go faster or slower and you still rely on friction.