Cars travelling on a banked curve

For a level (flat) curved road all of the centripetal force, acting on vehicles, must be provided by friction.

How can a car travel around a bend in the road when the surface is slippery or the car’s tyres have little tread?

Some curves are banked to compensate for slippery conditions like ice on a highway or oil on a racetrack.

Without friction, the roadway still exerts a normal force n perpendicular to its surface. And the downward force of the weight w is present.

Those two forces add as vectors to provide a resultant or net force F net which points toward the center of the circle; this is the centripetal force.

Note that it points to the center of the circle; it is not parallel to the banked roadway.

We can resolve the weight and normal forces into their horizontal and vertical components.

Since there is no acceleration in the y-direction so the sum of the forces in the y-direction must be zero. ie ncos  = mg

ie F nety = n cos  - w = 0 n cos  = w n = w / cos  n = mg / cos 

and F netx = n sin  F c = m v 2 / r but F c = F netx

F c = mv 2 / r = n sin  = [w / cos  ] sin  therefore F c = mv 2 / r = w [ sin  / cos  ] ie F c = w tan  m v 2 / r = m g tan  tan  = v 2 / r g Would a bank of angle  provide enough centripetal force for vehicles of all masses travelling at legal speeds around a bend in the road? Explain.