CIRCULAR MOTIONCIRCULAR MOTION

Angular MotionAngular Motion

• Angular displacement: • Angular velocity: • Angular acceleration • Uniformly accelerated motion

dtd

dtd

Linear Vs Angular KinematicsLinear Vs Angular KinematicsLinear Motion Angular Motion Relation-

ship(r = radius)

Quantity Unit Quantity Unit

s m rad s = r

v m s-1 rad s-1 v = r

a m s-2 rad s-2 a = r

• Period: T• Frequency: f

fT

22

Relation between Tangential and Relation between Tangential and Angular VelocitiesAngular Velocities

rv

Uniform Circular MotionUniform Circular Motion

• Tangential acceleration:

• Centripetal (Normal) acceleration:

0)1(coslim0

t

vatt

22

rrv

Centripetal ForceCentripetal Force

• A resultant force acting towards the centre• Centripetal acceleration• Centripetal force:

22

mrr

mvF

ConclusionConclusion

• Not a new type of force• Force velocity• Centripetal force does not imply the object will

move to the centre of the circle• Experimental verification• The force does no work on the object• If the force ceases to act, the object will move of

f tangentially

Experimental VerificationExperimental Verification

Computer simulation

Examples of Circular MotionExamples of Circular Motion

• Orbital motion of satellites and heavenly bodies

• Spinning of machine parts or wheels• Motion of charged particles in a magnetic

field• Early models of atoms

Further ExamplesFurther Examples

• Turning of a vehicle round a corner• Bicycle turning in a smooth banked track• Liquid spinning in a bucket about a vertical

axis• Aircraft turning in flight

Conical PendulumConical Pendulum

glT cos2 Period

Motion of Cyclist Round Circular Motion of Cyclist Round Circular TrackTrack• Condition for skidding:

tan > is independent of m• In turning a sharp

corner, must be large

Motion of Car round Circular Motion of Car round Circular TrackTrack

)(21 2

1 rahvgmR

)(21 2

2 rahvgmR

•Car will overturn if

•Car will skid if

hgarv

grv

BankingBanking

• For no side-slip at the wheels

• Daily example: racing track

grv2

tan

Aircraft Turning in FlightAircraft Turning in Flight

• Banking angle for the turn:

grv2

tan

CentrifugeCentrifuge

• To separate particles in suspension from the less dense liquid

• Procedure

RotorRotor

• The person will not slip down if

rg

Variation of g with LatitudeVariation of g with Latitude

• g’ = g - r2

Motion in a Vertical CircleMotion in a Vertical Circle

• Ring threaded on a smooth vertical circular wire [Figure]

• Suspended particle in a vertical circle [Figure]

• The outside of a smooth vertical circular rod [Figure]

Conditions of Describing a Conditions of Describing a Complete Vertical CircleComplete Vertical Circle• Case I: the particle is suspended by a light ri

gid rod

• Case II: the particle is suspended by a light string[Figure]

glv 20

glv 50

Bucket of Water Whirled in a Bucket of Water Whirled in a Vertical CircleVertical Circle• For the water to stay in the bucket: grv

Looping the loopLooping the loop

• To describe a complete circle:h 5r/2

ExamplesExamples

OrbitsOrbits

Back

Turning Round a CornerTurning Round a Corner

Centripetal force is provided by the frictional force between the wheels and the road

Back

Banked Track in CyclingBanked Track in Cycling

Centripetal force is provided by the horizontal component of the normal reaction.

Back

Ring Threaded on a Smooth Ring Threaded on a Smooth Vertical Circular WireVertical Circular Wire

Back

Suspended Particle in a Vertical Suspended Particle in a Vertical CircleCircle

Back

The Outside of a Smooth Vertical The Outside of a Smooth Vertical Circular RodCircular Rod

Back

Conditions for Describing a Conditions for Describing a Complete Vertical CircleComplete Vertical Circle

Back