Simple harmonic motion (SHM)

• When an object is undergoing SHM, it oscillates about an equilibrium point. It moves back and forth around that point, with changing velocity and changing acceleration.

  The equilibrium point is where there's no force. (straight down for pendulum or where spring rests). force on object always point back to equilibrium and gets bigger further away from equilibrium.

  
• Force that points to equilibrium is called the restoring force -> because it always tries to restore system to equilibrium.

  
  

  Equation:

  F = -ky, F is restoring force, k is a constant, y is displacement from equilibrium point of object (+ for up - for down)

  
  

  k depends on the system, if spring based then its a spring constant, lager spring constant means system is tighter and has bigger forces -> quicker oscillation

  If displacement gets bigger force does too in opposite direction. Force and displacement are proportional.
• Oscillations


  An oscillation is one back-and-forward cycle. We can call simple harmonic motion

  oscillatory motion
• Frequency of SHM

  The frequency is how many oscillations (full cycles) happen in one second. This doesn’t change with time, meaning the frequency at one time is the same as at another time, i.e. the frequency doesn’t slow down or speed up. As usual, the frequency has unit Hertz (Hz) and symbol f.
• Angular frequency

  The angular frequency has symbol ω and unit rad s−1.

  
  

  This is the same symbol as angular velocity in the rotational stuff because both rotate or oscillate at a certain frequency, and this is a way to measure that frequency. Better answer: often when rotational speed is constant we give it the special name angular frequency
• Period


  How long it takes for a full oscillation (cycle) to happen. Measured in seconds.

  T = 1/f
• Amplitude

  This is the size of the maximum displacement of the mass during its oscillation. This

  maximum is the same in both directions because SHM is symmetrical.
• Displacement is how far the mass is from the equilibrium, can be negative because it’s a vector.

  
  

  Equation: y = Asin (ωt), y is displacement (usually vertical or horizontal), A is amplitude of SHM, ω is angular frequency of SHM
• Velocity:  v = Aωcos (ωt)
• Acceleration: a = −Aω$^2$sin (ωt)

  Acceleration in terms of displacement: a = -ω$^2$y

  This all assumes that the displacement starts at zero (i.e. at the equilibrium point)

  when time is zero

  The displacement (y), velocity (v) and acceleration (a) of a mass undergoing SHM

  are all either sin or cos graphs. There are two sets of equations to use:

  Sin, cos, sin is for when you start (t = 0) at the equilibrium.

  Cos, sin, cos is for when you start (t = 0) at a maximum

  
• A phasor diagram is a unit circle with arrows drawn on it from the centre of the circle to the circumference. These show the movement of an SHM system at a given time.

  
• Mass on a spring, Spring force = restoring force

  
  

  Equation only for spring: T = 2π√mk

  If you have the period but want the angular frequency: ω = 2π/f= 2πT.
• Pendulum

  
  

  Length of pendulum is I and the mass attached to end of string is m

  
• Showing forces on pendulum that make it undergo SHM

  Two forces acting Fg and Ft.

  (on diagram Fg is -mg) in diagram mg is drawn as sum of two right angled vectors, breaking it up into its components: mgsin(θ) and mgcos(θ), -mgsin(θ) is the component force that is a restoring force (pulls pendulum back to equilibrium)

  
  

  Small angle approximation -> θ and sin(θ) pretty much equal. So replace sin(θ) with θ restoring force is roughy -mg(θ) so proportional to angular displacement and negative.

  force is proportional to linear displacement (y)

  
  

  Equation for period: T = 2π √ l/g

  
• Energy in SHM

  Energy is always conserved. If it appears that it’s not, then energy is leaving your system and going into the surroundings. The total energy in SHM stays the same.

  When there's no friction, total energy stays constant

  
  

  Equation: E = Ek + Ep, E is total energy in SHM system, Ek is kinetic energy of mass, Ep is potential energy in system.

  
  

  Potential energy for pendulum system: Ep = mgh

  Potential energy for spring mass system: Ep = 1/2 ky$^2$

  kinetic energy: 1/2mv$^2$

  
• Damped SHM

  Spring mass system when oscillating collides with air molecules losing a bit of energy with each collision

  
  

  when energy leaves system due to air friction, total energy must decrease overtime, causing amplitude to decrease overtime.

  
• Key points

  • Only EP(no EK) when the mass is at maximum displacement (also zero speed, maximum force and acceleration).
  • Only EK(no EP) when the mass is passing through at equilibrium (also maximum speed, zero force and acceleration).
  • Total E, given by EK+ EP stays constant (i.e. is a flat line) unless energy is leaving the system, e.g. through friction (in which case it is a steadily falling line
• Forced SHM


  Systems in normal SHM are operating at a natural frequency (resonant frequency), system wants to operate at this frequency. Applying force to system at natural frequency means energy is consistently being transferred this will increase amplitude of SHM over time as more energy is being transferred into the system.

  When the frequency of an applied force doesn’t match, it’s called a forced frequency. Such frequencies result in lower amplitudes and lower total energies