Banked Curves (Ideal Angle)
Overview
A banked road allows a vehicle to turn without friction. At the ideal banking angle θ for speed v: horizontal component of N provides centripetal force; vertical component balances weight. N cosθ = mg (vertical). N sinθ = mv²/r (centripetal). Dividing: tanθ = v²/(rg). At this angle, no friction needed. Above ideal speed: friction acts inward. Below ideal speed: friction acts outward. Exam tip: derive by resolving N into components — do NOT try to memorise; derive it every time.
Banked curves — how banking helps
On a flat road, friction provides centripetal force — if the road is icy (μ ≈ 0), cars cannot turn. Banked roads are tilted so the normal force N has a horizontal component pointing inward: N sin θ = mv²/r. The vertical component supports the weight: N cos θ = mg. Dividing: tan θ = v²/(rg). At this specific speed, no friction is needed. At other speeds, friction must provide the additional centripetal force.
- ⚠Using sin θ instead of tan θ in the formula tan θ = v²/(rg)
- ⚠Forgetting that the formula is only valid for the 'ideal' frictionless speed