Cars travelling on a banked curve

1. Cars travelling on a banked curve

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

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

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

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

6. 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.

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

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

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

10. ie Fnety = n cosq - w = 0 n cosq = w n = w / cosq n = mg / cosq

11. and Fnetx = n sinq Fc = m v2 / r but Fc = Fnetx

12. Fc = mv 2 / r = n sinq = [w / cosq ] sinq therefore Fc = mv 2 / r = w [ sinq / cosq] ie Fc = w tanq m v 2 / r = m g tanq tanq = v 2 / r g Would a bank of angle q provide enough centripetal force for vehicles of all masses travelling at legal speeds around a bend in the road? Explain.