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Circular Motion

- PHYA 4
- Further Mechanics

Many objects follow circular motion

- The hammer swung by a hammer thrower
- Clothes being dried in a spin drier
- Chemicals being separated in a centrifuge
- Cornering in a car or on a bike
- A stone being whirled round on a string
- A plane looping the loop
- A DVD, CD or record spinning on its turntable
- Satellites moving in orbits around the Earth
- A planet orbiting the Sun (almost circular orbit

for many) - Many fairground rides
- An electron in orbit about a nucleus
- So it is fairly common, and the maths is not too

hard!

How can we make an object travel in a circle?

- Hint think about Newtons 1st law...

Circular motion

- Remember Newtons 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 Newtons 2nd law?
- Fma
- 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!)

Circular motion

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

So the speed an object moves depends on the

frequency of rotation and the radius

Talking about circular motion

- Angular displacement (q) no. of radians turned

through - Angular speed (w) no. of radians turned through

per second - (sometimes called angular velocity)

Worked example Calculating w

- 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?

Worked example Calculating w

- A stone on a string the stone moves round at a

constant speed of 4 ms-1 on a string of length

0.75 m - What is the linear velocity of the stone at any

point on the circle? - Linear velocity of stone at any point on the

circle is3 ms1 directed along a tangent to the

point. - Note that although the magnitude of the linear

velocity (i.e. the speed) is constant its

direction is constantly changing as the stone

moves round the circle. - What is the angular velocity of stone at any

point on the circle? - Angular velocity of stone at any point on the

circle 3 /0.75 4 rad s1

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)

vector

Centripetal acceleration

object

Centripetal Force

- Force acts towards the centre of the circle, not

outwards! - Not a special type of force

Examples of sources of centripetal force

Planetary orbits gravitation

Electron orbits electrostatic force on electron

Centrifuge contact force (reaction) at the walls

Gramophone needle the walls of the groove in the record

Car cornering friction between road and tyres

Car cornering on banked track component of normal reaction

Aircraft banking horizontal 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.

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 m s-1. - Calculate
- (a) the centripetal acceleration of the stone
- acceleration v2/r 42 / 0.75 21.4 ms2
- (b) the centripetal force acting on the stone.
- F ma mv2/r 0.5 ? 42 / 0.75 10.7 N
- Notice that this is a linear acceleration and

not an angular acceleration. The angular velocity

of the stone is constant and so there is no

angular acceleration.

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

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- 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

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

m 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 q?

Banked tracks speed for no sideways friction

- Resolving reaction force horizontally and

vertically - so

Speed at which a vehicle can travel around a

banked curve without steering

Wall of death Ball of death

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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 natures way of finding

equilibrium - A system in disequilibrium has been disturbed

through the addition of energy - It oscillates and sheds this energy to regain

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
- Oscillation is simply a by-product of a system

out of equilibrium trying to restore its

equilibrium, but it is this by-product that

produces the most interesting results.

Oscillations in nature

- Oscillation is natures 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!
- Lets look at some examples...

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- 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

Equation for SHM

- Remember that restoring force was proportional to

displacement from equilibrium position F a x - F ma, so a a x or a w2x
- But a dv/dt d2x/dt2, so d2x/dt2 w2x
- Guess a solution x A sin(wt)
- dx/dt wA coswt,d2x/dt2 w2A sinwt w2x
- So it works, and we have derived expressions for

x, v and a (and hence F)

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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

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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.

Applications of damping

- Vehicle suspension
- Millennium bridge
- Auditorium acoustics
- Engine mounts
- Meter readouts
- Vibration isolation

Whats going on here?

- Example 2

Some good materials here

- www.acoustics.salford.ac.uk/feschools/index.htm

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 Bartons pendulums

- The amplitude of the forced oscillations depend

on the forcing frequency of the driver and reach

a maximum when forcing frequency natural

frequency of the driven cones. - The amplitude depends on the degree of damping,

(see graph below). - If damping is light, the frequency response curve

peaks sharply at the resonance frequency, and the

amplitude at resonance is very large. (See graph

below.) - If damping is heavy, the frequency response curve

is broader, and the amplitude at resonance is not

so large. - Once transient oscillations of varying amplitude

have died away a driven oscillator oscillates at

the forcing frequency. - At resonance the driver is one quarter of a cycle

(p /2) ahead of the driven oscillator (swing pic

p. 20) - If fnat lt fdriver then driver and driven are

nearly in antiphase. - If fnat gt fdriver then driver and driven are

nearly in phase.

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

Applications of resonance

- Musical instruments (strings, pipes, sound

boards) - Electrical circuits (eg radio tuner, filter)
- NMR imaging
- Laser cavities

Further investigation

- Pendulum lab
- Masses on springs