Oscillations

• An oscillation is a periodic to-and-fro motion of an object between two limits.
• The three types of oscillations include:

  • Free oscillation
  • Damped oscillation
  • Forced oscillation
• When an object undergoes free oscillation, it oscillates with no energy gain or loss. The object oscillates with constant amplitude, as there are no external force acting on it.
• Examples of free oscillations include:

  • A simple vertical spring-mass system
  • A simple horizontal spring-mass system
  • A simple pendulum
• Amplitude is the maximum displacement of the oscillating object from the equilibrium position.
• Displacement is the distance of the oscillating object from its equilibrium position in a stated direction.
• Period is the time taken for one complete oscillation.
• Frequency is the number of complete to-and-fro cycles per unit time made by the oscillating object.
• $f =$$\frac{1}{T}$ (in name)


  f = Frequency

  T = Period
• Angular frequency is the constant which characterises the particular simple harmonic oscillator and is related to the natural frequency given by 2πf.
• $\omega=$$\frac{2\pi}{T}=$$2\pi{f}$ (in name)


  $\omega$ = Angular Frequency

  T = Period

  f = Frequency
• Phase is the angle which gives a measure of the fraction of a cycle that has been completed by an oscillating particle or a wave.
• Phase difference between two oscillations is a measure of how much one oscillation is out of step with another.
• When one oscillation reaches a positive maximum displacement as another oscillation reaches a negative maximum displacement, both oscillations are said to be in anti-phase.
• $x =$$x_0sin(\omega{t} +$$\phi)$ (in name)


  $x$ = Displacement

  $x_0$ = Amplitude

  $\omega$ = Angular Frequency

  t = Time

  $\phi$ = Phase
• Simple harmonic motion is defined as the oscillatory motion of a particle whose acceleration is directly proportional to its displacement from a fixed point and is always opposite to the direction of displacement.
• $a =$$-\omega^2{x}$ (in name)


  a = Acceleration

  $\omega$ = Angular Frequency

  x = Displacement
• $v =$$v_0{\space}cos\omega{t}$ (in name)


  v = Velocity

  $v_0$ = Maximum Velocity

  $\omega$ = Angular Frequency

  t = Time
• $v_0=$$x_0\omega$ (in name)


  $v_0$ = Maximum Velocity

  $x_0$ = Amplitude

  $\omega$ = Angular Frequency
• $a=$$-a_0{\space}sin\omega{t}$ (in name)


  a = Acceleration

  $a_0$ = Maximum Acceleration

  $\omega$ = Angular Frequency

  t = Time
• $a_0 =$$x_0\omega^2$ (in name)


  $a_0$ = Maximum Acceleration

  $x_0$ = Amplitude

  $\omega$ = Angular Frequency
• $v =$$\pm\omega\sqrt{(x_0^2-x^2)}$ (in name)


  v = Velocity

  $\omega$ = Angular Frequency

  $x_0$ = Amplitude

  x = Displacement
• If (a) is the graph of displacement against time, what is (b)?

  Velocity against Time

  
• If (a) is the graph of displacement against time, what is (b)?

  Acceleration against Time

  
• If (a) is the graph of velocity against time, what is (b)?

  Acceleration against Time

  
• If (a) is the graph of velocity against time, what is (b)?

  Displacement against Time

  
• The maximum kinetic energy of an oscillator in SHM is given by:

  $E_{Kmax} =$$\frac{1}{2}\omega^2x_0^2$, where (in name)


  $\omega$ = Angular Frequency

  $x_0$ = Amplitude
• When x = 0 at t = 0,

  A) Potential Energy

  B) Kinetic Energy

  C) Total Energy

  
• For an oscillator in simple harmonic motion, the Energy-Time graphs represents:

  $\frac{1}{2}m\omega^2x_0^2$ = Total Energy

  $\frac{1}{2}m\omega^2x_0^2cos^2\omega{t}$ = Kinetic Energy

  $\frac{1}{2}m\omega^2x_0^2sin^2\omega{t}$ = Potential Energy
• For a pendulum bob,

  A) Total Energy

  B) Potential Energy

  C) Kinetic Energy

  
• Input the correct terms:

  A) y

  B) y = -y0

  C) y = 0

  D) y = +y0

  E) KE

  F) GPE

  G) EPE

  H) TE

  
• Damping is a process where energy is taken from an oscillating system as a result of dissipative forces.
• Damped oscillation occurs when there is a continuous dissipation of energy to the surroundings such that the total energy in the system decreases with time, hence the amplitude of the motion progressively decreases with time.
• Input the correct degrees of damping:

  A) Light Damping

  B) Heavy Damping

  C) Critical Damping

  
• Applications of damping:

  • Car Suspension System
  • Analog Voltmeters and Ammeters
• Forced oscillations are caused by the continual input of energy by external applied force to an oscillating system to compensate the loss due to damping in order to maintain the amplitude of the oscillation.
• Resonance occurs when the resulting amplitude of the system becomes a maximum when the driving frequency of external driving force equals to the natural frequency of the system.
• In a Frequency Response Graph, as damping increases,

  • The response is less sharp
  • The maximum amplitude is reached when the driving frequency is slightly less than the natural frequency