# Oscillations

- 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

- Mass on A Spring
- 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

- Resonance happens when driving frequency = natural frequency

- The frequency of this force is called the driving frequency
- 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

- Forced
oscillations
involve adding
energy to a
system whilst
it oscillates
- 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

- The amount
of damping
will change
how quickly
the amplitude
is reduced
- 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

- Lightly damped systems have a very sharp resonance peak

- Simple Harmonic Motion
- 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)

- When the object is moving away from the
midpoint, all that KE is transferred back into PE.

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