Acceleration● Recall that when a nett force acts on an objectit causes an acceleration.● This is the basis of Newtons second law● This acceleration will cause a change invelocity of the object● It will change speed,● It will change direction● It will change speed and direction
Centripetal Acceleration● Consider an object that is forced to go in acircle by a string tied at the centre of that circle.● IF the string is cut, the object will move off witha linear velocity.● IF the object is moving at constant speed then;● There can be no component of acceleration in thedirection of the velocity.● There must still be an acceleration towards thecentre of the circle because the string is there.● The direction of the velocity must change by thisradial acceleration
Centripetal Acceleration● Acceleration is the rateof change of velocity.● Assume that the objectmoves from A to B intime t.● The change in velocitycan be foundgraphically by v – uuvrrθuvsθ
Centripetal Acceleration● Consider that thedisplacement triangleand the velocity triangleare mathematicallysimilar if the speed andradius of the circle areconstant.uvrrθuv vv=srsθ
Centripetal Acceleration● The average velocity is definedas:Substituting this in the previousequation gives:● Which rearranges to:uvrrθuvv=s tsθ vv=v tr v t=v2r=ac
Centripetal Acceleration● The object travels through a total angle of2π radians in one orbit in a time Period ofT● The angular speed is therefore● The angle turned in time t is therefore(distance = speed x time):● Degrees can be converted into radians by:uvrrθuvsθ=2Trad =deg360×2= t
Centripetal Acceleration● Using the angularspeed the orbital speedis thus:● Which makes theacceleration equation:uvrrθuvsθv=2 rT=r ac=r 2
Centripetal Acceleration● Mathematically, the position vector of a particle attime t is given by:● Differentiating for v gives:● Differentiating again gives the acceleration:r=rcos xr sin yr=rcos t xr sin t yv=ddtr =−r sin t xr cos t ya=ddtv=d2dt2r=−r 2cos t x−r 2sin t ya=−2r cost xr sin t ya=−2rxyrrsinθr cos θ
What causes this Acceleration?● As has been shown, the uniform circularacceleration occurs along a radius (radially) asshown by the r in the equation and towards thecentre of the circle, as shown by the -.● Any force that acts in this way will cause acentripetal acceleration.● e.g. the gravitational pull of the Earth on the Mooncauses the acceleration that keeps the Moon inorbit.● The sideways friction force (towards the centre) thatkeeps a cars tyres from slipping outwards causesthe acceleration to make the car turn the corner.
Centrifugal Acceleration● A centrifugal (centre-fleeing) acceleration doesnot really exist.● It is a consequence of Newtons first law thatstates that objects with inertia will continue in astraight line unless an external force acts.● When we go around a corner in a car, we feel thatwe are being thrown outwards.● In reality our inertia causes us to try to go straight.● The forces acting on the car tyres makes the caraccelerate (turn) across us.● We collide with the side of the car, which exerts asideways force on us (Newtons 3rd), that causes us
Questions● The Moon orbits the Earth at a radius of3.2x106m in a time period of 28.5 days. What isthe centripetal acceleration of the Moon?● The Earth orbits the Sun at a radius of 150million km in a year. What is the centripetalacceleration of the Earth?● A formula one car travels around a bend ofradius 25m at 110kmh-1. What is its centripetalacceleration?
Questions● The tyres of a car of mass 1500kg can exert amaximum sideways force of 500N whencornering. What is the maximum speed thatthis car can travel around a bend of radius 8mand radius 24m?● A formula 1 car has a mass of 625kg. Thedriver wants to take a corner at no less than90kmh-1using tyres that can exert a maximumsideways force of 35kN. What is the minimumradius corner that the car can cope with beforeslipping?