Circular Motion

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Circular Motion

• PHYA 4
• Further Mechanics

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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!

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How can we make an object travel in a circle?

• Hint think about Newtons 1st law...

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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!

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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?

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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!)

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Circular motion
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Talking about circular motion

• The radian

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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
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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)

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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?

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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

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Practice Questions

• Examples 1 Radians and angular speed

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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
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Centripetal acceleration
object
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Centripetal Force

• Force acts towards the centre of the circle, not
  outwards!
• Not a special type of force

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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
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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.

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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.

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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

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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...

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Practice Questions

• Centripetal force sheet
• Whirling bung experiment
• Examples sheet 2

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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...
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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)
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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)

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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?

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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

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Simple Harmonic Motion

• PHYA 4
• Further Mechanics

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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.

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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...

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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

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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?

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Displacement of mass on a spring
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Mass on spring terminology
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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

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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
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Mass on spring Energy
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SHM is like a 1D projection of uniform circular
motion
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Phasors

• A rotating vector which represents a wave
• Length corresponds to amplitude, angle
  corresponds to phase

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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
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Damping example
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Under-damping
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Critical Damping

• Critical damping provides the quickest approach
  to zero amplitude

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Over-damping
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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.

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Applications of damping

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

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Whats going on here?

• Example 2

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Some good materials here

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

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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

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• 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.

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Resonant driving
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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.

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Amplitude vs driving frequency
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Effect of damping on resonance
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Applications of resonance

• Musical instruments (strings, pipes, sound
  boards)
• Electrical circuits (eg radio tuner, filter)
• NMR imaging
• Laser cavities

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Further investigation

• Pendulum lab
• Masses on springs