topic 13 oscillations

• Free and Forced Oscillations

  • An object oscillates or vibrates when it moves back and forth repeatedly on either side of some fixed position (equilibrium position)
  • Free oscillations occur without a driving mechanism and the oscillating object continues to move for some time after being set into oscillation
• Examples of oscillating objects

  • Thermal vibrations of atoms in a solid, swaying motion of the top of a skyscraper, plucked guitar string, skin of a banged drum, beating of a hummingbird's wings, x-rays, microwaves, radio waves
• When an object is set into free oscillation, it will vibrate at a particular frequency called the natural frequency of vibration
• A forced oscillation occurs when an object is caused to vibrate by a periodic driving force, making the object vibrate at the forcing frequency
• Examples of forced oscillations


  • Engine vibrations felt in the steering wheel and gear stick of a car
• The natural frequency of vibration of an oscillator is the frequency with which it will vibrate freely after an initial disturbance
• The wing beat of a mosquito, the pendulum movement in a Grandfather clock, and the vibrations of a cymbal after it has been struck are examples of free oscillations
• The shaking of a building during an earthquake and the vibration of a bat are examples of forced oscillations
• Types of oscillations


  • Free oscillations
  • Forced oscillations
• Free oscillations


  • The pendulum movement in a Grandfather clock
  • The vibrations of a cymbal after it has been struck
  • The vibration of a bat after a cricket ball is struck
• Forced oscillations


  • The shaking of a building during an earthquake
  • The vibrations of a washing machine during its spin cycle
  • Vibrating loudspeaker cone
• Heavily-loaded trolley attached by identical springs

  When pulled horizontally to one side and released, it oscillates freely back and forth along the bench
• Speed of the trolley during oscillation

  Greatest at the centre, zero at the extremities
• Kinetic and potential energy during oscillation of the trolley

  Maximum kinetic energy and minimum potential energy at the centre, maximum potential energy and zero kinetic energy at the extremities
• Pendulum bob oscillation


  Freely oscillates at its natural frequency
• Speed of the pendulum bob during oscillation

  Maximum at the centre, zero at the extremities
• Kinetic and potential energy during pendulum oscillation

  Maximum kinetic energy at the centre, zero at the extremities; minimum potential energy at the centre, maximum at the extremities
• Examples of SHM


  • Swinging pendulum, oscillating mass-spring system, vibrating loudspeaker cone, vibrations of atoms or molecules in a solid
• The equation for a SINUSOIDAL oscillation is: x = A sinωt = A sin(2πft) OR x = A cos(2πft)
• MAXIMUM SPEED (vmax) of a simple harmonic oscillator

  The maximum speed of a simple harmonic oscillator
• Displacement (x)


  Distance moved by an oscillating object in either direction from the equilibrium position at any given time
• Amplitude (A)


  Maximum displacement of an oscillating object from the equilibrium position
• Period (T)


  Time taken for each complete oscillation
• Frequency (f)


  Number of complete oscillations per second
• Angular Frequency (ω)


  Frequency of the oscillations expressed in radians per second
• Oscillations and circular motion are closely related
• For 1 complete revolution of P, which is 1 complete oscillation of the foot of its perpendicular across XY: ω = 2π/T and ω = 2πf (since T = 1/f)
• PHASE is the term used to describe the point that an oscillating object has reached within the complete cycle of an oscillation
• PHASE DIFFERENCE between two oscillations tells us the amount by which they are 'out of step' (out of phase) with each other
• In Phase


  Two points that have exactly the same pattern of oscillation
• Antiphase


  If the patterns of movement at two points are exactly opposite to each other
• Simple Harmonic Motion (SHM)

  The oscillatory motion of an object in which acceleration is directly proportional to its displacement from a fixed point and always in the opposite direction to the displacement
• Maximum Displacement


  xmax = ±A (where A = amplitude)
• Time Period (T)


  Independent of the amplitude of the oscillations
• The variation of displacement (x) with time (t) depends on its initial displacement
• If x = 0 when t = 0, the displacement at time (t) is given by x = A sin(2πft)
• If x = +A when t = 0, the displacement at time (t) is given by x = A cos(2πft)
• The v/t graph can be deduced from the x/t graph because v = dx/dt (velocity = gradient of the displacement/time graph)
• v is +ve when dx/dt is +ve (object moving to the right) and v is -ve when dx/dt is -ve (object moving to the left)
• Simple Harmonic Oscillations


  1. Displacement (x)

  2. Acceleration (a)

  3. Velocity (v)