Further Mechanics

• Angular speed

  • angle an object rotates through per second
  • ω = θ/t
  • v = ωr
  • ω = angular speed (rads^-1)
  • θ = angle object turns through (rad)
  • t = time (s)
  • v = linear velocity (ms^-1)
  • r = radius of circle of rotation (m)
• Frequency, f (revs^-1, Hz)

  • number of complete revolutions per second

  Period, T (s)

  • time taken for a complete revolution

  ω = $\frac{2\pi}{T}$=$2\pi f$
• Centripetal acceleration

  • directed towards centre of the circle
  • a = $\frac{v^2}{r}$=$ω^2r$
• Centripetal force

  • a resultant force which acts towards the centre of the circle
  • is what keeps the object moving in a circle, remove the force and object would fly off at a tangent with velocity v
  • F = $\frac{mv^2}{r}$ (=$mω^2r$)
• Simple Harmonic Motion

  • an oscillation in which the acceleration of an object is directly proportional to its displacement from its equilibrium position, and directed towards the equilibrium
  • a ∝ -x
  • minus sign shows acceleration is always opposing the displacement (directed towards the equilibrium)
• Displacement

  • $x=$$Acos(ωt)$

  Acceleration

  • $a=$$-ω^2x$
  • $a_{max}=$$ω^2A$

  Velocity

  • $v=$$\pmω\sqrt{A^2-x^2}$
  • $v_{max}=$$ωA$
• Potential and Kinetic Energy

  • an object in SHM exchanges potential and kinetic energy as it oscillates
  • the type of potential energy depends on the restoring force - gravitational for pendulums, elastic and sometimes gravitational for springs
  • as object moves towards the equilibrium, restoring force does work on object and transfers EP -> EK until equilibrium reached and EK is a maximum and EP is zero
  • as object moves away from equilibrium, all the EK -> EP until max displacement where EP is maximum and EK is zero
  • sum of EP and EK is constant (without damping)
• Potential and Kinetic Energy

  A) Total

  B) Potential

  C) Kinetic

  D) amplitude

  E) equilibrium

  F) constant

  
• Mass on a Spring

  • is a simple harmonic oscillator. When the mass is pushed or pulled either side of the equilibrium position, there's a restoring force exerted on it.
  • an ideal oscillator would undergo free oscillations where there are no energy losses over the cycle
  • The size and direction of this restoring force is given by Hooke's Law:
  • F = kΔL
  • Period of a mass oscillating on a spring:
  • $T=$$2\pi\sqrt{\frac{m}{k}}$​​
  • T = period (s)
  • m = mass (kg)
  • k = spring constant (Nm^-1)
• The Simple Pendulum

  • Period of an oscillating pendulum:
  • $T=$$2\pi\sqrt{\frac{l}{g}}$
  • T = period (s)
  • l = length of pendulum (m)
  • g = gravitational field strength (Nkg-1)

  It only works for small angles.
• Free Vibrations

  • involve no transfer of energy to or from the surroundings
  • oscillates at its natural frequency, what it will naturally vibrate at when allowed to oscillate freely

  Forced Vibrations

  • when a system is forced to vibrate by a periodic external force
  • the frequency of this force is the driving frequency
  • if driving frequency is much less than natural frequency the two are in phase, if much larger then they will be in antiphase
• Resonance

  • 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
  • For system with no damping, when driving frequency = natural frequency, the system is resonating
  • at resonance the phase difference between driver and oscillator is 90°
• Damping

  • a reduction in the amplitude of an oscillation as a result of energy being lost from the system to overcome frictional or other resistive forces
• Light Damping

  • amplitude of the system gradually decreases, reducing by the same fraction each cycle
  • exponential decay of amplitude with time

  Critical Damping

  • system returns to equilibrium in the shortest time after displacement

  Heavy/Overdamping

  • so strong the oscillator returns to equilibrium much more slowly than when the object is critically damped

  A) Displacement

  B) Light

  C) Critical

  D) Heavy

  
• Damping and resonance peaks

  • frequency of max amplitude is the resonant frequency
  • the more damped a system:
  • the smaller max amplitude
  • the further resonant frequency is from natural frequency
  • the more broad the resonant peak

  A) increasing

  B) light

  C) heavier

  D) resonance peak

  E) increasing

  F) natural

  G) resonant

  H) less

  I) heavier

  J) driving

  K) amplitude