further mechanics

• how does lighter damping affect resonance?

  the maximum amplitude at resonance becomes larger and the resonant frequency becomes closer to the natural frequency of the system, this makes the resonance curve sharper and steeper (if damping isn't light, resonance occurs at a slightly lower frequency than the natural frequency, has a lower amplitude, and the peak of the amplitude becomes wider)

  
• what are some examples of resonance?

  - pushing someone on a swing

  - buildings in an earthquake

  - army on a bridge

  - wind on a bridge

  - wine glass being smashed

  - hair cells in the cochlea

  - Barton's pendulums
• what is a resonance curve?

  
• what is resonance?

  when an object vibrates at its natural frequency its rate of energy transfer is at a maximum, which increases the amplitude greatly, and the phase difference between the driver and the oscillations becomes π / 2 radians

  
  

  for a system with little or no damping, at resonance,

  applied frequency of the periodic force = natural frequency of the system
• what is a periodic force?


  a force applied at regular intervals
• what happens as the frequency of forced oscillations on a system increases?

  the amplitude of the systems oscillations increases until it reaches maximum amplitude at it's natural frequency and then amplitude decreases again
• what is a forced oscillation?

  an oscillation produced by an external periodic force that oscillates at the driving frequency produced by an external force called the driver, if the driving frequency equals the natural frequency of the system then resonance occurs
• what is overdamping?


  a type of damped oscillation where the resistance is increased beyond critical damping so the object takes longer to return to equilibrium, no oscillating motion occurs in this, examples of this include closing doors and fancy toilet seats (these work by using oil filled pistons)
• what is critical damping?

  a type of damped oscillation where the resistance is such that the oscillation returns to equilibrium in the shortest time possible without oscillating, an example of this is in car suspension systems
• what is heavy damping?


  a type of damped oscillation where each cycle takes the same length of time but there is increased resistance, which means the amplitude decreases faster than in light damping, dying away faster, an example of this is a pendulum with a card attached
• what is light damping?

  a type of damped oscillation where time period is independent of amplitude so each cycle takes the same length of time but the amplitude reduces exponentially over time, eventually dying away, examples of light damping are pendulums and mass on a spring
• what are dissipative forces?

  forces that dissipate the energy of a system to the surroundings as thermal energy (friction and air resistance decreasing amplitude and therefore energy in an oscillating pendulum)
• what is a damped oscillation?

  an oscillation where energy is dissipated to the surroundings due to friction and air resistance, this decreases the amplitude and causes the oscillations to die away eventually
• what is a free oscillation?

  an oscillation with constant amplitude as no friction acts on it so it oscillates forever at the same amplitude, so there is no energy transferred to or from the surroundings
• what is the energy-displacement graph?

  
• what is the small angle approximation for cos(x)?

  cos(x) = 1 - ((x^2) / 2)
• what is the small angle approximation for sin(x)?

  sin(x) = x
• what is the SHM speed equation and how is it derived?


  v = +/- ω x √(A^2 - x^2)

  
  

  Ek = Et - Ep

  1/2mv^2 = 1/2kA^2 - 1/2kx^2

  = v^2 = (k / m) x (A^2 - x^2)

  as ω^2 = k / m, v = +/- ω x √(A^2 - x^2)

  
  

  v = velocity (m s^-1)

  ω = angular frequency (rad s^-1)

  A = amplitude (m)

  x = displacement (m)
• what is the equation for total energy?

  Et = 1/2 x k x A^2

  
  

  Et = total energy (J)

  k = spring constant (N m^-1)

  A = amplitude (m)
• what is the equation for potential energy of a mass on a spring?

  Ep = 1/2 x k x x^2

  
  

  Ep = potential energy (J)

  k = spring constant (N m^-1)

  x = displacement (m)
• at what point is an object in SHM at maximum acceleration and max potential energy?

  at maximum displacement (amplitude)
• at what point is an object in SHM at maximum velocity and max kinetic energy?


  at equilibrium position
• what are the uses for the equation for time period of a mass on a spring?

  - to measure the mass of objects in space on the ISS where they would otherwise appear as weightless as it is independent of gravity

  
  

  - mechanical watches - the period is independent of the initial displacement, so the mass on a spring keeps a regular period as the oscillations decrease in amplitude over time
• what is the equation for time period of a mass on a spring, the restoring force, and how is it derived?

  T = 2π x √(m / k)

  
  

  using Hooke's Law (F = k x ΔL), change of tension, ΔT = -kX, ∴ the restoring force = -kX

  
  

  so the acceleration = - (kX) / m (as F = m x a, so -kX = m x a), which through subbing in the SHM equation we get ω^2 = k / m, which is (2π / T)^2 = k / m, this leads to the equation T = 2π x √(m / k)

  
  

  T = time period (s)

  m = mass on spring (kg)

  k = spring constant (N m^-1)

  ΔL = extension (m)

  X = displacement (m)
• what determines the frequency of oscillation of a loaded spring?

  using a trolley attached to two stretched springs

  
  

  - changing the extra mass - more mass increases the inertia of the system, so at a given displacement the trolley would be slower than without the extra mass, so each oscillation takes longer

  
  

  - using weaker springs - restoring force on trolley at any displacement would be less, so the trolley's acceleration and speed at any given displacement would be less, so each oscillation would take longer
• what are some historical uses of pendulums?

  - timing - for small displacement angles, a pendulum keeps a constant time even as the amplitude of its oscillations decrease so it can be used for timing, as seen in grandfather clocks

  
  

  - measuring strength of gravity - as time period for a pendulum depends on g, a long pendulum of known length was historically used to measure variation of g in differing locations by measuring time for a large number of oscillations
• what happens when a pendulum is displaced through angles larger than 10°?


  the force and acceleration returning to the equilibrium position don't increase with displacement (conditions for SHM are not met), this is because T = 2π x √(L / g) uses a small angle approximation, which becomes invalid for angles larger than 10°
• what are the weight components, the restoring force, and sinθ of a simple pendulum?


  m x g x cosθ = the component of weight perpendicular to the path of the bob

  m x g x sinθ = the component of weight along the path towards the equilibrium position

  
  

  ∴ the restoring force = -m x g x sinθ, so acceleration = -g x sinθ

  
  

  provided θ doesn't exceed 10° sinθ = s / L, so a = -(g / L) x s = -ω^2 x s

  
  

  m = mass of bob (kg)

  g = strength of gravity (m s^-2 or n kg^-1)

  θ = angle to the vertical

  l = length of the thread from pivot to object (m)

  s = displacement from equilibrium position (m)
• what is the pendulum equation and two important take-aways from it?


  T = 2π x √(L / g)

  
  

  T = time period (s)

  L = length (m)

  g = strength of gravity (m s^-2 or n kg^-1)

  
  

  - T isn't affected by the object's mass

  - T isn't affected by the magnitude of displacement
• what is the resultant force for oscillating objects?

  the force that acts towards the equilibrium position (this is known as the restoring force), which is proportional to the displacement from equilibrium, for the object to oscillate with SHM
• what is the equation for maximum acceleration?


  amax = +- ω^2 x A

  
  

  amax = maximum acceleration (m s-2)

  ω = angular frequency (rad s-1)

  A = amplitude (m)
• what is the equation for maximum velocity?


  vmax = +- ω x A

  
  

  vmax = maximum velocity (m s-1)

  ω = angular frequency (rad s-1)

  A = amplitude (m)
• what are some examples of SHM?

  - pendulum (<10°)

  - mass on a spring

  - tides
• what does SHM look like on an acceleration/displacement graph?

  
• what does SHM look like on an acceleration-time graph?

  an upside-down cos graph for the x = A x cos(ω x t) variant

  
• what does SHM look like on a displacement-time graph?

  a cos graph for the x = A x cos(ω x t) variant

  
• what does SHM look like on a velocity-time graph?

  an upside-down sin graph for the x = A x cos(ω x t) variant

  
• what are the two displacement equations from the SHM equation?

  x = A x sin(ω x t), when x = 0 at t = 0

  
  

  x = A x cos(ω x t), when x = A at t = 0

  
  

  A = amplitude (m)

  ω = angular frequency (rad s-1)

  x = displacement (m)

  t = time (s)
• what is simple harmonic motion (SHM) and its equations?

  the motion of an object about a fixed point where the acceleration is proportional to the displacement and is directed towards the fixed point

  
  

  a ∝ -x

  a = -ω^2 * x

  a = -((2 x π) / T)^2 * x

  a = -(2 x π x f)^2 * x

  
  

  a = acceleration (m s-2)

  ω = angular frequency (rad s-1)

  x = displacement (m)
• what is phase difference and its equation?

  in radians, for two objects oscillating with the same T, phase difference = (2 x π x Δt) / T

  
  

  Δt = time between successive instants when the two objects are at max displacement in the same direction

  
  

  if the two objects oscillate in phase, Δt = T