1. Centripetal Force & the Road

2. Centripetal Force & The Road
Learning Objectives
Book Reference : Pages 26-27
To show that the centripetal force is provided by real world forces such as tension, gravity & friction
To consider three particular cases of motion:
Over the top of a hill or humped back bridge
Around flat curves (roundabouts)
Around banked curves

3. Centripetal Acceleration : Recap
During the last lesson we saw that an object moving in a circle has a constantly changing velocity, it is therefore experiencing acceleration and hence a force towards the centre of rotation.
We called this the centripetal force: The force required to keep the object moving in a circle. In reality this force is provided by another force, e.g. The tension in a string, friction or the force of gravity.

4. Over the top 1
Consider a car with mass m and speed v moving over the top of a hill... S mg r

5. Over the top 2
At the top of the hill, the support force S, is in the opposite direction to the weight (mg). It is the resultant between these two forces which keep the car moving in a circle
mg – S = mv2 / r
If the speed of the car increases, there will eventually be a speed v0 where the car will leave the ground (the support force S is 0)
mg = mv02 / r v0 = (gr)½
Any faster and the car will leave the ground

6. Around a Roundabout 1
On a level road, when a car travels around a roundabout the centripetal force required to keep the car moving in a circle is provided by the friction between the road surface and tyres
velocity
friction
Force of Friction F
F = mv2 / r

7. Around a Roundabout 2
To avoid skidding or slipping, the force of friction F0 must be less than the point where friction is overcome which occurs at speed v0
Friction is proportional to weight and can be given by the coefficient of friction ():
F  mg F = mg
At the point of slipping:
F0 = mv02 / r  mg = mv02 / r
 v0 = (gr)½

8. Banked Tracks 1
For high speed travel, race tracks etc have banked corners. In this way a component of the car’s weight is helping friction keep the car moving in a circle N1 N2  
Towards centre of rotation mg 

9. Banked Tracks 2
Without any banking the centripetal force is provided by friction alone. Banked corners allows greater speeds before friction is overcome
The centripetal force is provided by the horizontal components of the support forces
(N1 + N2) sin  = mv2 / r
and the vertical components balance the weight
(N1 + N2) cos  = mg

10. Banked Tracks 3 Rearranging sin  = mv2 / (N1 + N2) r
cos  = mg / (N1 + N2)
and since tan  = sin  / cos 
tan  = mv2 / (N1 + N2) r x (N1 + N2) / mg
tan  = mv2 / mgr v2 = gr tan 
Thus there is no sideways frictional force if the speed v is such that v2 = gr tan 

11. Problems 1
A car with mass 1200kg passes over a bridge with a radius of curvature of 15m at a speed of 10 ms-1. Calculate:
The centripetal acceleration of the car on the bridge
The support force on the car when it is at the top
The maximum speed without skidding for a car with mass 750kg on a roundabout of radius 20m is 9ms-1. Calculate:
The centripetal acceleration of the car on the roundabout
The centripetal force at this speed

12. Problems 2
A car is racing on a track banked at 25° to the horizontal on a bend with radius of curvature of 350m
Show that the maximum speed at which the car can take the bend without sideways friction is 40ms-1
Explain what will happen if the car takes the bend at ever increasing speeds