Circular Motion

PHYA 4

Further Mechanics

How can we make an object travel in a circle?

• Hint: think about Newton’s 1st law...

Circular motion• Remember Newton’s 1st law?

– an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force

• So what is needed to make something go around in a circle?– A resultant force

• Remember Newton’s 2nd law?– F=ma

• So a body travelling in a circle constantly experiences a resultant force (and is accelerated) towards the centre of the circle– This is not an equilibrium situation! An unbalanced

force exists!

A bucket of water on a rope

• If we spin the bucket fast enough in a vertical circle, the water stays in the bucket– Why?

A mass on a string

• Speed of rotation remains constant• Velocity is constantly changing, so mass is

constantly accelerating towards centre of circle• So there is a constant force on the mass

towards the centre of the circle– Tension in string (until you let go!)

Talking about circular motion

The radian

Rotation and speed

• No gears, so as the pedals are turned, the wheel goes round with them with a period T

• The wheel rim is travelling faster than the pedals, although both are rotating at the same frequency, f

• Speed of rim: rfT

rv

22

So the speed an

object moves depends on the frequency of

rotation and the radius

s

r

Talking about circular motion

• Angular displacement = no. of radians turned through

• Angular speed () = no. of radians turned through per second

(sometimes called angular velocity)

rvr

v

t

rs

dt

dT

f

or ,/

22 speed

Worked example: Calculating

A stone on a string: the stone moves round at a constant speed of 3 ms-1 on a string of length 0.75 m

•What is the instantaneous linear speed of the stone at any point on the circle?

•What is the angular speed of stone at any point on the circle?

Practice Questions

• Examples 1: Radians and angular speed

Centripetal acceleration

• Acceleration directed towards centre– Centripetal means “centre seeking”

• Size depends on:– How sharply the object is turning (r)– How quickly the object is moving (v)

dt

da

v vector

Centripetal acceleration

r

vra

rv

vat

t

v

t

va

vvv

v

r

s

22

,

.

so

Remember

so

,but

so ,

object

Centripetal Force

• Force acts towards the centre of the circle, not outwards!

• Not a special type of force

rmr

mvF

maF

22

,

Examples of sources of centripetal force

Planetary orbits gravitation

Electron orbits electrostatic force on electron

Centrifugecontact force (reaction) at the walls

Gramophone needlethe walls of the groove in the record

Car cornering friction between road and tyres

Car cornering on banked track

component of normal reaction

Aircraft bankinghorizontal component of lift on the wings

Worked Example: Centripetal Force

A stone of mass 0.5 kg is swung round in a horizontal circle (on a frictionless surface) of radius 0.75 m with a steady speed of 4 ms-1.

Calculate:

(a) the centripetal acceleration of the stone

(b) the centripetal force acting on the stone.

No such thing as centrifugal force...

• Centrifugal means “centre fleeing”

• It is an “effective force” you feel when in a rotating frame of reference

• e.g., cornering car

No such thing as centrifugal force...

• Car applies a force towards the centre of the circle

• Driver feels a force pushing him outwards– Reaction force

• Physics joke...

Practice Questions

• Centripetal force sheet

• Whirling bung experiment

• Examples sheet 2

Hump-backed bridges

• Centripetal force provided by gravity

• Above a certain speed, v0, this force is not enough to keep vehicle in contact with road

gr vso 0

20 r

mvmg Note: independent of mass...

Roundabouts and corners

• What provides the centriptal force?– Friction

• What factors affect the maximum speed a vehicle can corner?– Radius of corner– Limiting frictional force

r

mvF

20

0 mg

: coefficient of friction (not examinable)

Banked tracks• On a flat road, only friction

provides the centripetal force– Above a certain speed you

lose grip

• On a banked track there is a horizontal component of the reaction force towards the centre of the curve– No need to steer! (at least at

one particular speed)

Optimum speed on a banked track

• Can you derive an expression for the speed at which no steering is required for a circular track of radius r, banked at an angle

Banked tracks – speed for no sideways friction

• Resolving reaction force horizontally and vertically:

• so

mgnr

mvn

cos

sin2

tantan 22

grvgr

v or ,

Speed at which a vehicle can travel around a banked curve without steering

Wall of death Ball of death

Fairgrounds• Many rides derive their excitement

from centripetal force– A popular context for exam questions!

– Read pages 26-29– Answer questions on p.29

Simple Harmonic Motion

PHYA 4

Further Mechanics

Oscillations in nature• Oscillation is nature’s way of finding equilibrium• This interplay can be found throughout nature:

– A swinging pendulum– Waves on water– A plucked string (and the eardrum of a listener)– Vibrating atoms in a lattice– Voltages and currents in electric circuits– Excited electrons emitting light– A bouncing ball– Ocean tides– Populations of predators and prey in an ecosystem...

Simple Harmonic Motion

• Harmonic motion: motion that repeats itself after a cycle

• Simple: simple!

• Let’s look at some examples...

time

disp

lace

men

t

time

time

velo

city

acce

lera

tion

• Displacement/velocity/acceleration animation

• x/v/a Java applet

Simple Harmonic Motion Summary

• What is SHM?

• What sort of systems display SHM?

• How can we describe SHM?

• What is happening to the energy of an ideal system undergoing SHM?

Displacement of mass on a spring

Mass on spring terminology

When do you get SHM?

• A system is said to oscillate with SHM if the restoring force:– is proportional to the displacement from

equilibrium position– is always directed towards the equilbrium

position

Force, acceleration, velocity and displacement

If this is how the displacement varieswith time...

... the velocity is the rate of changeof displacement...

... the acceleration is the rate ofchange of velocity...

...and the acceleration tracks the forceexactly...

... the force is exactly opposite tothe displacement...

Phase differences Time traces varies with time like:

/2 = 90

/2 = 90

= 180

zero

displacement s

force F = –ks

displacement s

cos 2ft

same thing

–sin 2ft

–cos 2ft

–cos 2ft

cos 2ft

acceleration = F/m

velocity v

Mass on spring Energy transfer

Mass on spring Energy

SHM is like a 1D projection of uniform circular motion

Phasors

• A rotating vector which represents a wave

• Length corresponds to amplitude, angle corresponds to phase

Damping• In a real system there is always some

energy loss to the surroundings

• This leads to a gradual decrease in the amplitude of the oscillation– For light damping, the period is

(approximately) unaffected, though.

• The damping force generally is linearly proportional to velocity– Resulting in exponential decrease of

amplitude

Damping

Damping example

Under-damping

Critical Damping

• Critical damping provides the quickest approach to zero amplitude

Over-damping

Damping summary

• An underdamped oscillator approaches zero quickly, but overshoots and oscillates around it

• A critically damped oscillator has the quickest approach to zero.

• An overdamped oscillator approaches zero more slowly.

What’s going on here?

• Example 2

Free and Forced vibration

• When a system is displaced from its equilibrium position it oscillates freely at its natural frequency– No external force acts– No energy is transferred

• When an external force is repeatedly applied the system undergoes forced oscillation– energy is transferred to the system.

• Eg Barton’s pendulums

Resonant driving

Resonance

• If the system happens to be driven at its natural frequency the transfer of energy is most efficient: this is RESONANCE– Oscillation is positively reinforced every cycle– Amplitude quickly builds up

• Resonance can lead to uncontrolled, destructive vibrations– Bridges, glasses and opera singers, etc.

Amplitude vs driving frequency

Effect of damping on resonance

Further investigation

• Pendulum lab

• Masses on springs