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Circular Motion

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Circular Motion PHYA 4 Further Mechanics Amplitude vs driving frequency Effect of damping on resonance Applications of resonance Musical instruments (strings, pipes ... – PowerPoint PPT presentation

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Title: Circular Motion


1
Circular Motion
  • PHYA 4
  • Further Mechanics

2
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!

3
How can we make an object travel in a circle?
  • Hint think about Newtons 1st law...

4
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!

5
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?

6
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!)

7
Circular motion
8
Talking about circular motion
  • The radian

9
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
10
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)

11
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?

12
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

13
Practice Questions
  • Examples 1 Radians and angular speed

14
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
15
Centripetal acceleration
object
16
Centripetal Force
  • Force acts towards the centre of the circle, not
    outwards!
  • Not a special type of force

17
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
18
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.

19
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.

20
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

21
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

22
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23
  • Physics joke...

24
Practice Questions
  • Centripetal force sheet
  • Whirling bung experiment
  • Examples sheet 2

25
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...
26
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)
27
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)

28
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?

29
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
30
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31
Fairgrounds
  • Many rides derive their excitement from
    centripetal force
  • A popular context for exam questions!
  • Read pages 26-29
  • Answer questions on p.29

32
Simple Harmonic Motion
  • PHYA 4
  • Further Mechanics

33
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.

34
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...

35
Simple Harmonic Motion
  • Harmonic motion motion that repeats itself after
    a cycle
  • Simple simple!
  • Lets look at some examples...

36
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37
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38
  • Displacement/velocity/acceleration animation
  • x/v/a Java applet

39
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?

40
Displacement of mass on a spring
41
Mass on spring terminology
42
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

43
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)

44
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45
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46
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47
Mass on spring Energy transfer
48
Mass on spring Energy
49
SHM is like a 1D projection of uniform circular
motion
50
Phasors
  • A rotating vector which represents a wave
  • Length corresponds to amplitude, angle
    corresponds to phase

51
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

52
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53
Damping
54
Damping example
55
Under-damping
56
Critical Damping
  • Critical damping provides the quickest approach
    to zero amplitude

57
Over-damping
58
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.

59
Applications of damping
  • Vehicle suspension
  • Millennium bridge
  • Auditorium acoustics
  • Engine mounts
  • Meter readouts
  • Vibration isolation

60
Whats going on here?
  • Example 2

61
Some good materials here
  • www.acoustics.salford.ac.uk/feschools/index.htm 

62
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

63
  • 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.

64
Resonant driving
65
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.

66
Amplitude vs driving frequency
67
Effect of damping on resonance
68
Applications of resonance
  • Musical instruments (strings, pipes, sound
    boards)
  • Electrical circuits (eg radio tuner, filter)
  • NMR imaging
  • Laser cavities

69
Further investigation
  • Pendulum lab
  • Masses on springs
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