A banked curve is a turn where the road surface is tilted so the outside edge is higher than the inside edge. This design helps cars and bikes change direction without relying only on friction between the tires and the road. Banked curves matter because every object moving in a circle needs an inward centripetal force.
By tilting the surface, engineers use part of the normal force to point toward the center of the curve.
Key Facts
- Centripetal force requirement: F_c = mv^2/r
- On a frictionless ideal banked curve: tan θ = v^2/(rg)
- Ideal speed on a banked curve: v = sqrt(rg tan θ)
- The horizontal component of the normal force helps provide centripetal force: N sin θ = mv^2/r
- The vertical component of the normal force balances weight in the ideal frictionless case: N cos θ = mg
- Racetracks are banked so cars can turn at higher speeds with less dependence on tire friction.
Vocabulary
- Banked curve
- A curved path whose surface is tilted so the normal force has an inward component.
- Centripetal force
- The net inward force required to keep an object moving in a circular path.
- Normal force
- The contact force exerted by a surface perpendicular to that surface.
- Banking angle
- The angle θ between the tilted road surface and the horizontal.
- Ideal speed
- The speed at which a vehicle can take a banked curve without needing friction for the inward force.
Common Mistakes to Avoid
- Pointing the normal force straight up is wrong because the normal force is always perpendicular to the sloped road surface.
- Using F_c as an extra force is wrong because centripetal force is the name for the net inward force, not a separate force added to the diagram.
- Forgetting the radius in tan θ = v^2/(rg) is wrong because a tighter curve requires a larger banking angle at the same speed.
- Assuming friction is always necessary on a banked curve is wrong because at the ideal speed, the inward component of the normal force can provide the needed centripetal force by itself.
Practice Questions
- 1 A frictionless banked curve has radius 50 m and banking angle 20°. What is the ideal speed for a car on the curve? Use g = 9.8 m/s^2.
- 2 A highway curve is designed for an ideal speed of 25 m/s and has radius 120 m. What banking angle should it have? Use g = 9.8 m/s^2.
- 3 A car drives around a banked curve slower than the ideal speed. Explain which way friction would act if friction is present and why.