1.1. To show that the centripetal force is To show that the centripetal force is provided by real world forces such as provided by real world forces such as tension, gravity & frictiontension, gravity & friction

2.2. To consider three particular cases of motion:To consider three particular cases of motion:

• Over the top of a hill or humped back Over the top of a hill or humped back bridgebridge

• Around flat curves (roundabouts)Around flat curves (roundabouts)

• Around banked curvesAround banked curves

Book Reference : Pages 26-27Book Reference : Pages 26-27

During the last lesson we saw that an object During the last lesson we saw that an object moving in a circle has a constantly changing moving in a circle has a constantly changing velocity, it is therefore experiencing acceleration velocity, it is therefore experiencing acceleration and hence a force towards the centre of and hence a force towards the centre of rotation.rotation.

We called this the centripetal force: The force We called this the centripetal force: The force required to keep the object moving in a circle. In required to keep the object moving in a circle. In reality this force is provided by another force, reality this force is provided by another force, e.g. The tension in a string, friction or the force e.g. The tension in a string, friction or the force of gravity.of gravity.

Consider a car with mass Consider a car with mass mm and speed and speed vv moving moving over the top of a hill...over the top of a hill...

mg

r

S

At the top of the hill, the support force S, is in the At the top of the hill, the support force S, is in the opposite direction to the weight (mg). It is the opposite direction to the weight (mg). It is the resultant between these two forces which keep resultant between these two forces which keep the car moving in a circlethe car moving in a circle

mg – S = mvmg – S = mv22 / r / r

If the speed of the car increases, there will If the speed of the car increases, there will eventually be a speed eventually be a speed vv00 where the car will leave where the car will leave the ground (the support force S is 0)the ground (the support force S is 0)

mg = mvmg = mv0022 / r / r vv00 = (gr) = (gr)½½

Any faster and the car will leave the groundAny faster and the car will leave the ground

On a On a levellevel road, when a car travels around a road, when a car travels around a roundabout the centripetal force required to keep roundabout the centripetal force required to keep the car moving in a circle is provided by the the car moving in a circle is provided by the friction between the road surface and tyresfriction between the road surface and tyres

velocityfriction

Force of Friction FForce of Friction F

F = mvF = mv22 / r / r

To avoid skidding or slipping, the force of friction To avoid skidding or slipping, the force of friction FF00 must be less than the point where friction is must be less than the point where friction is overcome which occurs at speed overcome which occurs at speed vv00

Friction is proportional to weight and can be given Friction is proportional to weight and can be given by the coefficient of friction (by the coefficient of friction ():):

F F mg mg F = F = mgmg

At the point of slipping:At the point of slipping:

FF00 = mv = mv0022 / r / r mg = mvmg = mv00

22 / r / r

vv00 = ( = (gr)gr)½½

For high speed travel, race tracks etc have banked For high speed travel, race tracks etc have banked corners. In this way a component of the car’s corners. In this way a component of the car’s weight is helping friction keep the car moving in a weight is helping friction keep the car moving in a circlecircle

mg

N1

N2

Towards centre of rotation

Without any banking the centripetal force is Without any banking the centripetal force is provided by friction alone. Banked corners allows provided by friction alone. Banked corners allows greater speeds before friction is overcomegreater speeds before friction is overcome

The centripetal force is provided by the horizontal The centripetal force is provided by the horizontal components of the support forcescomponents of the support forces

(N(N11 + N + N22) sin ) sin = mv = mv22 / r / r

and the vertical components balance the weightand the vertical components balance the weight

(N(N11 + N + N22) cos ) cos = mg = mg

RearrangingRearranging

sin sin = mv = mv22 / (N / (N11 + N + N22) r) r

cos cos = mg / (N = mg / (N11 + N + N22))

and since tan and since tan = sin = sin / cos / cos

tan tan = mv = mv22 / (N / (N11 + N + N22) r x (N) r x (N11 + N + N22) / mg) / mg

tan tan = mv = mv22 / mgr / mgr vv2 2 = gr tan = gr tan

Thus there is no sideways frictional force if Thus there is no sideways frictional force if the speed v is such that the speed v is such that vv2 2 = gr tan = gr tan

A car with mass 1200kg passes over a bridge with a radius A car with mass 1200kg passes over a bridge with a radius of curvature of 15m at a speed of 10 msof curvature of 15m at a speed of 10 ms-1-1. Calculate:. Calculate:

a.a. The centripetal acceleration of the car on the bridgeThe centripetal acceleration of the car on the bridge

b.b. The support force on the car when it is at the topThe support force on the car when it is at the top

The maximum speed without skidding for a car with The maximum speed without skidding for a car with mass 750kg on a roundabout of radius 20m is 9msmass 750kg on a roundabout of radius 20m is 9ms-1-1. . Calculate:Calculate:

a.a. The centripetal acceleration of the car on the The centripetal acceleration of the car on the roundaboutroundabout

b.b. The centripetal force at this speedThe centripetal force at this speed

A car is racing on a track banked at 25° to the A car is racing on a track banked at 25° to the horizontal on a bend with radius of curvature of horizontal on a bend with radius of curvature of 350m350m

a.a. Show that the maximum speed at which the Show that the maximum speed at which the car can take the bend without sideways car can take the bend without sideways friction is 40msfriction is 40ms-1-1

b.b. Explain what will happen if the car takes the Explain what will happen if the car takes the bend at ever increasing speedsbend at ever increasing speeds