Oscillations

Simple Harmonic Motion
1. OSCILLATE
  1. To 'OSCILLATE' is to undertake continuously repeated movements.
    1. E.g. a person on a swing.
    2. E.g. a child on a pogo-stick.
    3. E.g. a pendulum in a Grandfather clock.
2. PERIOD
  1. A 'PERIOD' is the time taken to complete one complete oscillation.
3. SIMPLE HARMONIC MOTION
  1. 'SHM' is when a system is oscillating such that a force is trying to return the object to its centre position and this force is proportional to the distance from that centre position.
  2. F = -kx
    1. F = Restoring Force (N)
    2. k = constant(N / m)
    3. x = distance from equilibrium (m)
4. The SHM definition, is the same as that expressed by Hooke's Law. As a result, we can find the equation for the period of oscillations of a mass, subject to F by:
  1. T = √(m÷k)
    1. T = Time period (s)
    2. m = mass (kg)
    3. k = constant (N/m)
5. Famous physicist Galileo was 17 when he observed the period of an oscillation in a cathedral. He then came up with another equation:
  1. T = 2π√(l÷g)
    1. T = Time period (s)
    2. l = length of pendulum
    3. g = gravitational field strength (m/s/s)
  2. PENDULUMS
SHM Mathematics
1. 1. ω= θ/t
    1. ω= angular velocity (rad/s)
    2. θ = angle (radians)
    3. t = time (s)
    4. As ϴ = ωt, then ϴ can be replaced with ωt .
  2. From this graph, you could calculate 'x' from doing x = rCosϴ
    1. SO: x = A cos ωt
2. If you combine the two equations below, you can find ω by doing ω= 2πf
  1. T = 1/f
    1. T = time period (s)
    2. f = frequency (1/s)
  2. T = 2π / f
    1. T = time period (s)
    2. f = frequency (1/s)
3. 1. x = A cos ωt
    1. DIFFERENTIATE
  2. v = -Aω^2 sin ωt
    1. DIFFERENTIATE
  3. a = -Aω^2 cos ωt
    1. a = - ω^2 x
    2. As a = -kx / m, the constant k can be found with k = ω^2 m
SHM Energy
1. With reference to the 'Energy of Conservation' law, an oscillator cannot gain or lose energy, unless there is an external force.
2. The idea that an oscillator cannot lose/gain energy can be represented as a graph.
  2. The constant transfer of energy, would be enough to keep any oscillator in constant motion, however, we know that in reality resistance forces act to stop this. This could be air resistance or resistance between two materials.
Resonance and Damping
1. FREE OSCILLATION
  1. 'FREE OSCILLATION' happens when a system performs oscillations, free oscillations of any forces from outside of the system.
    1. E.g. releasing a pendulum.
2. NATURAL FREQUENCY
  1. A 'NATRUAL FREQUENCY' is the frequency of oscillations that a system will take if it undergoes free oscillations.
3. FORCED OSCILLATIONS
  1. A 'FORCED OSCILLATION' is when a system oscillates under the influence of an external force. (NOT SHM)
  2. This involves adding energy to a system.
4. DRIVING FREQUENCY
  1. A 'DRIVING FREQUENCY' is the frequency of an external force applied to the system undergoing FORCED OSCILLATIONS.
5. RESONANCE
  1. 'RESONANCE' describes very large amplitude oscillations that occur when a DRIVING FREQUENCY matches the NATURAL FREQUENCY.
6. 1. DAMPING
    1. 'DAMPING' is the material, or system, causing energy loss during each DAMPED OSCILLATION.
  2. DAMPED OSCILLATIONS
    1. 'DAMPED OSCILLATIONS' occur with each oscillation; there is a loss in energy and this reduces the amplitude over time.
  3. CRITICAL DAMPING
    1. A 'CRITICALLY DAMPED' system occurs when damping is such that the oscillator returns to its equilibrium position in the quickest possible time, without going past it.
  4. Dampers are vital to modern day society. They are used in cars, and earthquake prone buildings.

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Oscillations

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