a level physics SHM

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a level physics SHM
2014-10-13 14:30:27
simple harmoinic motion

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  1. why do old clocks use a pendulum
    a pendulum oscillates backwards and forwards with a regular beat .
  2. even as the oscillation dies away the time period
    stays the same because . this is because the amplitude of the swing also decreases so the pendulum moves a smaller distance at a smaller speed hence the time period stays the same
  3. SHM
    • simple harmonic motion 
    • when an object oscillates with a constant time period even if the amplitude varies we say it's moving with SHM . 
    • also in SHM acceleration is proportional to the displacement of the object from equilibrium and is always directed towards the equilibrium position 
    • so in SHM acceleration acts in the opposite direction to displacement
  4. velocity on a pendulum/swing
    • at each end of the oscillation you are stationary for a moment 
    • the pendulum/swing speeds up as you move back to the centre 
    • once you pass this point you slow down again 
    • velocity is a vector quantity so we need to consider its direction we can take forward motion as positive and backward motion as negative
  5. acceleration on a swing
    • starting at 1 you are accelerating towards 3 (the equilibrium position) 
    • once you pass 3 you slow down until you stop at 5 . deceleration is the same as negative acceleration so decelerating away from 3 is the same as accelerating towards 3 . hence there's still acceleration towards the centre . 
    • from 5 you accelerate back towards the centre again 
    • from 3 to 1 you decelerate to a stop but this is the same as accelerating towards 1 - the centre 
    • throughout the motion acceleration is directed towards the centre (equilibrium position)
  6. how does the size of acceleration change
    • when you pull the mass to one side and let it go it oscillates backwards and forwards 
    • when the mass is displaced to the left tension in k1 decreases and tension in k2 increases so the resultant pull of the strings is back to the right which which pulls the mass back to the centre (equilibrium position) each time 
    • newtons second law tells us the greater the force the greater the acceleration so the greater the force the greater the acceleration so the greater the displacement from the equilibrium position the greater the acceleration
  7. how can we produce a displacement time graph for a simple pendulum
    • using a pendulum pen 
    • as the pen swings back an forth it draws over the same line on the paper underneath 
    • if we pull the paper along steadily beneath the pen the pen would draw a regular wave 
    • this is the shape of the displacement time graph for SHM
  8. The displacement time graph for SHM is
    • sinusoidal . it is the shape of a sine or cosine curve
    • displacement is a vector quantity so displacement in one direction is taken as positive and those in the opposite direction are taken as negative
  9. one complete oscillation means 
    • a movement from one extreme to the other and back again 
    • the time this takes is called the time period , T
    • the number of oscillations per second is called the frequency , f
  10. frequency is measured in hertz
  11. the amplitude of the motion is
    the maximum displacement from the equilibrium position
  12. velocity is the rate of
    change of displacement
  13. the velocity at any point in SHM is equal to the
    gradient of the displacement time graph
  14. if we started timing at the equilibrium position for a displacement time graph the gradient is maximum when
    • displacement is zero , the gradient falls to zero as the swing reaches its maximum amplitude 
    • this tells us that the velocity is greatest when the displacement is zero and falls to zero at the extremes
  15. acceleration is
    the rate of change of velocity
  16. the acceleration at any point in SHM is
    the gradient of the velocity time graph
  17. if we started timing at the equilibrium position for a velocity time graph the gradient is maximum when
    the velocity is zero this is at maximum amplitude . so the acceleration is greatest at maximum amplitude , maximum amplitude is maximum displacement so acceleration increases as displacement increases
  18. when the displacement is positive
    acceleration is negative
  19. if we start timing at the equilibrium position the displacement time graph is
    velocity time graph is 
    acceleration time graph is
  20. ignore side equations
  21. we use cosine if the
    timing starts at max displacement
  22. we use sine if timing starts at
    the centre of oscillation
  23. displacement --> velocity ---> acceleration
    • sin --> - cos --> -sin
    • cos --> -sin --> -cos
  24. using circular motion amplitude =
  25. using circular motion if we start at max displacement 
    x =
  26. w =2pif 
    w = angular velocity 
    w =
    • velocity = distance / time 
    • w = θ/t 
    • wt = θ
  27. so x =
  28. wt =
  29. so x =
  30. f = 1/t so x =
  31. velocity = differentiation of displacement so if we started max displacement v =
    • v = -2pifAsin(2pift) 
    • v = -wAsin(wt)
    • v = -2pi/TAsin(2pit/T)
  32. for these sorts of calculations we need to have our calc in
    radians mode
  33. acceleration = differentiation of velocity so if we start at max displacement a =
    • a = -4pi2f2Acos(2pift) 
    • a = -w2Acos(wt) 
    • a = -4pi2/T2Acos(2pit/T)
    • Acos(2pift) = x so
    • a = -4pi2f2x  a = (2pif)2x
    • a = -w2x
    • a=-4pi2/T2
  34. oscillations could have
    different periods and amplitudes
  35. however oscillations can have the same period and amplitude but differ
    the oscillations could be out of step . to describe how far out of step oscillations are we use the idea of phase
  36. phase measures
    how far through a cycle the movement is
  37. one complete oscillation has a phase of
    2pi radias
  38. a pendulum oscillates with SHM if
    the amplitude is small
  39. the time period of pendulum depends on
    • length of the pendulum 
    • as the string gets longer the time period increases
  40. the time period of a pendulum is independent of the
    mass of the bob
  41. if the bob is displaced from equilibrium then released it
    oscillates about the lowest/highest point
  42. at displacement x from the highest/lowest point when the thread is at an angle θ to the vertical the mass has two components
    • mg=tcosθ vertically 
    • -ma = tsinθ
  43. restoring force = t
    • mg/cosθ = -ma/sinθ
    • mgsinθ/cosθ = -ma 
    • gtanθ= - a
  44. provided θ is small
    • gtanθ = -a
    • if θ small tanθ = sin θ 
    • sinθ=x/l 
    • gx/l = -a
  45. if θ greater than 10
    restoring force wouldn't be proportional to displacement from equilibrium position because other forces now need consideration
  46. a also = -(2pif)2x
    • -(2pif)2x = -gx/l 
    • -(2pif)2= g/l 
    • 2pif = √g/l 
    • f = √g/l / 2pi 
    • f= 1/T
    • T = 2pi√l/g
  47. measuring acceleration due to gravity
    T = 2pi√l/g
    • measure T with a stopwatch 
    • draw a graph of Tvs l 
    • T2 = 4pi2l/g 
    • T2 = 4pi2/g * l
    • y = m * x +c 
    • line of best fit should be a straight line through origin as no c so no y intercept 
    • m = 4pi2/g
  48. if we hang a mass on a spring and pull it down a small way and let it go the mass bounces up and down about its equilibrium it oscillates with sum what impacts the time period
    • mass and stiffness 
    • the greater the mass the slower it accelerates under the same force this means the time period will increase 
    • a stiffer spring will pull the mass back to its equilibrium with more force , this produces more acceleration if the mass moves faster the time period will decrease
  49. tension in spring = kΔl
    t = -kx --> negative because the tension acts upwards trying to restore the object to its equilibrium
  50. restoring force =
  51. f = ma
    • a = f/m 
    • a = -kx/m
  52. a also = -(2pif)2
    • -(2pif)2x=-kx/m 
    • -(2pif)2=k/m
    • 2pif=√k/m 
    • f = √k/m/2pi 
    • T = 1/f 
    • T = 2pi√m/k
  53. we can calculate k in
    a similar way to how we measured g for the simple pendulum the difference would be a graph of T2 vs mass
  54. frequency of a mass on a spring is increased if
    k is increased or m os reduced
  55. time period in space doesn't depend on
    g so to measure the mass of an astronaut they are placed between two springs and the time period is known since k is known we can calculate m
  56. tension in the spring varies from
    mg +kA to mg - KA where A = amplitude
  57. minimum tension is when the spring is
    compressed as much as possible when -A = x
  58. maximum tension occurs when
    the spring is stretched as much as possible when X = A
  59. how would you use a displacement time graph to show the motion is SHM
    show the variation is sinusoidal
  60. if T is the same for two different oscillations and the springs have the same constant
    the masses must be identical
  61. simple pendulum - at max amplitude you are
    stationary for a moment so kinetic energy is zero
  62. kinetic energy reaches a maximum
    at the centre as you speed up towards this point
  63. gravitational potential energy has a max value
    • at the extremes and lowest value at centre 
    • so in moving from an extreme to centre potential energy is transferred to kinetic energy
  64. mass oscillating on spring
    energy changes from kinetic to elastic potential energy and back again every half cycle after passing through equilibrium
  65. provided friction is constant , the total energy of the system is constant and is equal to
    max potential energy
  66. ep =
    • ep = 0.5fx
    • f = kx
    • ep = 0.5kx2
  67. ep max =
    • 0.5kA2
  68. change in ep =
  69. ep lost = ek gained 
    • 0.5mv2=0.5k(A2-X2
    • v2=k(A2-X2)/m 
    • √k/m = 2pif 
    • v = 2pif √A2-x2 
    • when x = 0 vmax = 2pifA
  70. the potential energy curve is
    parabolic in shape given by ep = 0.5kx2
  71. kinetic energy curve is an
    • inverted parabola given by Ek = 0.5k(A2-x2) since kinetic energy is total energy - potential energy 
    • 0.5kA2 - 0.5kx2
  72. the sum of the kinetic energy and the potential energy is
    always equal to 0.5kA2 which is the same as the kinetic energy at zero displacement so the two curves add together to give a straight line for total energy
  73. total energy =
    kinetic energy max = 0.5mvmax2 = 0.5m(2pifA)2 = 0.5m4pi2f2A=m2pi2f2A2
  74. total energy is therefor proportional to
  75. so far we have assumed that no energy is lost from an oscillating system and that it continues to oscillate indefinitely , this is known as
    free oscillations
  76. but if you leave a pendulum or mass on a spring oscillating it eventually slows down and stops , but
    the time period stays constant because both the amplitude and speed get smaller
  77. what slows the oscillations down
    air resistance
  78. energy is lost in
    overcoming air resistance this effect is known as damping
  79. light damping graph
  80. light damping is when
    oscillations take a while to die away
  81. heavier damping
  82. example of heavier damping
    imagine a pendulum moving through water , once released it would take longer to return to its equilibrium position and would hardly oscillate at all
  83. critical damping graph
  84. critical damping
    just enough to stop the system oscillating after it has been displaced and released from equilibrium
  85. over damping graph
  86. over damping
    it doesn't oscillate after it has been displaced and takes a while to return to the equilibrium position
  87. example of over damping
    a pendulum moving through thick treacle
  88. over damping is useful where
    rapid fluctuations need to be ignored , an example of this is a car fuel gauge , over damping stops the pointer oscillating as the fuel sloshes in the tank
  89. forced oscillations are
    oscillations of a system that's subjected to an external periodic force
  90. if you want to make a child's swing ho higher
    you push in time with the swings movement
  91. a periodic force is
    an applied force at regular intervals
  92. if you want amplitude to increase
    the frequency of the periodic force must match the natural frequency or resonant frequency of the oscillator
  93. driving frequency is
    • applied frequency
    • i.e. frequency of periodic force
  94. if driving frequency matches applied frequency then
    amplitude builds up energy is transferred from the driver to the oscillator .
  95. this effect is called
  96. the lighter the amplitude
    the larger the amplitude becomes
  97. resonance can be
  98. cartoons sometimes show glasses smashing when somebody hits a high note . this can really happen
    tje frequency of the sound (driving frequency) must match the natural frequency of the glass . the glass then resonates , vibrating more and more until it breaks
  99. the phase difference between the periodic force and displacement is always ....when resonance occurs in other words ...
    • pi/2 rad
    • the driver leads the resonator by pi/2 rad
  100. this means that
    the periodic force is exactly in phase with the velocity of the oscillating object
  101. as the applied frequency (driving frequency) increases towards the natural frequency
    the phase relationship between displacement and periodic force increases from 0 to pi/2 rad so from in phase to pi/2 rad out of phase
  102. as the applied frequency (driving frequency) increases past the natural frequency the phase relationship between displacement and periodic force
    increase from pi/2 rad out of phase to pi rad out of phase (so from out of phase to anti phase)
  103. when there's a phase difference
    the driver leads the resonator (oscillator)
  104. amplitude frequency graph
  105. below the natural frequency the amplitude of the oscillation will be
    below the amplitude of the driver
  106. at the natural frequency the amplitude of the oscillation
    will increase and increase
  107. above the natural frequency the amplitude of the oscillations
    will decrease more and more
  108. damping reduces the effect of
  109. without damping the
    amplitude increases until the object is under too much pressure so breaks
  110. as damping increases
    • amplitude of the resonance increases 
    • resonance peak gets broader 
    • resonant frequency is slightly lower than natural frequency
  111. damping is used where
    resonance can be a problem
  112. a good example is the damping of
    buildings in earthquake zones . the foundations are designed to absorb energy . this stops the amplitude of the buildings oscillations reaching dangerous levels when an earthquake arrives
  113. resonant frequency is only equal to
    natural frequency if the damping is minimal
  114. explain the formation of unpleasant noises coming from a loudspeaker
    • a loudspeaker vibrates in response to the oscillating electrical signal that drives it , thus it undergoes forced vibrations
    • forced vibrations occur at the driving frequency of electrical signals 
    • if the driving frequency matches the natural frequency of the loudspeaker then large amplitude vibrations occur
  115. tacoma narrows bridge
    frequency of the forces from the wind matched natural frequency of the bridge and large amplitude oscillations built up destroying the bridge a similar thing happened in frame however the frequency of the marching soldiers matched the natural frequency of the bridge
  116. how do microwaves work
    the frequency of microwaves almost equals natural frequency of vibration of water molecules . this means the water molecules in food resonate . this means they take in energy from the microwaves and get hotter , this heats the food , there is a slight mismatch in frequencies it prevents all the energy being absorbed at the surface and allows microwaves to penetrate deeper in the food
  117. what happens if x is displaced and released so it oscillates in the plane perpendicular to the plane of the pendulum at rest 
    the effect of the oscillating motion of x is transmitted along the support thread , subjecting each of the other pendulums to forced oscillations . Pendulum D will oscillates with the largest amplitude .Pendulum X and D have equal length and consequently equal natural frequency. Therefore resonance happens to pendulum D, and it oscillates with maximum amplitude.
  118. at any point in a progressive wave
    the amplitude and frequency are the same
  119. damping force is always in the opposite direction to ...
  120. damping reduces
    max kinetic and potential energy
  121. displacement and amplitude have
    two directions
  122. to travel from displacement to the centre of oscillation is
    1/4 of the time period
  123. acceleration is least when
    • speed is greatest
    • displacement is least
  124. acceleration is greatest when
    • amplitude is greatest 
    • velocity is least
  125. in sum the graph of acceleration vs displacement is
    a straight line
  126. time period of a pendulum on the moon would be .... than the time period of the same pendulum on earth
  127. To celebrate the Millennium in the year 2000, a footbridge was constructed across the River
    Thames in London. After the bridge was opened to the public it was discovered that the structure
    could easily be set into oscillation when large numbers of pedestrians were walking across it.
    (b) Under what condition would this phenomenon become particularly hazardous? Explain your answer.
    • driving force is at same frequency as natural frequency of structure 
    • resonance therefor occurs 
    • large amplitude vibrations produced 
    • could cause damage to structure
  128. Suggest two measures which engineers might adopt in order to reduce the size of the
    oscillations of a bridge
    • stiffen the structure 
    • increase damping by installing dampers