Oscillations

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  • Created by: Dani
  • Created on: 10-05-15 12:46
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  • Oscillations
    • Simple Harmonic Motion
      • Definition: an oscillation in which the acceleration is directly proportional to its displacement from the midpoint, and is directed towards the midpoint
      • The time taken to complete one cycle is called the time period (measured in seconds)
        • The number of cycles per second is the frequency (measured in Hz)
      • Restoring Force: the force trying to return the object to its centre position.
        • Proportional to the distance of that centre position.
        • F= -kx (can relate this to F=ma)
        • Energy
          • As the object moves towards the midpoint, the restoring force does work on the object and transfers some PE to KE
          • When the object is moving away from the midpoint, all that KE is transferred back into PE. 
            • As the object moves towards the midpoint, the restoring force does work on the object and transfers some PE to KE
          • At the midpoint KE=max and PE=0
            • At the amplitude (maximum displacement)  KE= 0 and PE= max
          • Potential Energy + Kinetic Energy = Mechanical Energy and is always constant (as long as the motion isn’t damped)
      • Graphs
        • Displacement x varies as a cosine or sine wave with a max value, A (amplitude)
        • Velocity: v is the gradient of the displacement-time graph. It has a max value of Aw (w= angular speed) and is a quarter of a cycle in front of the displacement
        • Acceleration: a is the gradient of the velocity-time graph. It has a max value of Aw^2 and is in antiphase with the displacement
      • Equations
        • w=    /t
        • a = - w ² x
        • a= -A w²cos(wt)
        • v=-Awsin(wt)
        • x= Acos(wt)
    • Simple Harmonic Oscillations
      • Mass on A Spring
        • F=-kx where k is the spring constant
        • Time period of a mass oscillating on a spring =    2
        • T²  m
        • T²  1/k
        • T does not depend on A
      • Pendulum
        • Time period of a mass oscillating on a pendulum = 2
        • T²  l
        • T does not depend on A
        • T does not depend on m
    • Forced and Free Oscillations
      • Forced oscillations involve adding energy to a system whilst it oscillates
        • The frequency of this force is called the driving frequency
          • Resonance happens when driving frequency = natural frequency
            • if no energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude for ever
            • when the driving frequency approaches the natural frequency, the system gains more and more energy from the driving force and so vibrates with a rapidly increasing amplitude. When this happens the system is resonating
      • Releasing a pendulum and letting it swing freely is a free oscillation
        • if no energy is transferred to or from the surroundings, it will keep oscillating with the same amplitude for ever
        • Any oscillating system has a natural By repeated frequency
      • Damping
        • Happens when energy is lost to the surroundings
        • Different amounts of damping have different effects
          • The amount of damping will change how quickly the amplitude is reduced
            • Light Damping
            • Heavy Damping
            • Critical Damping
            • Over Damping
        • Resonance
          • Lightly damped systems have a very sharp resonance peak
            • Amplitude only increases dramatically when the driving frequency is very close to the natural frequency
          • Heavily damped systems have a flatter resonance peak
            • The amplitude doesn't increase very much near the natural frequency and they aren't as sensitive to the driving frequency
    • Restoring Force: the force trying to return the object to its centre position.
      • Proportional to the distance of that centre position.
      • F= -kx (can relate this to F=ma)
      • Energy
        • When the object is moving away from the midpoint, all that KE is transferred back into PE. 
          • At the midpoint KE=max and PE=0
            • At the amplitude (maximum displacement)  KE= 0 and PE= max
          • Potential Energy + Kinetic Energy = Mechanical Energy and is always constant (as long as the motion isn’t damped)

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