# Oscillations

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• Created by: Dani
• Created on: 10-05-15 12:46
• 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)