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 Fnet 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 Fnety = n cos - w = 0 n cos = wn = w / cos

n = mg / cos

and Fnetx = n sin Fc = m v2 / r

butFc = Fnetx

Fc = mv 2 / r = n sin = [w / cos ] sin therefore Fc = mv 2 / r = w [ sin / cos]

ie Fc = 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.