Describing Oscillations

• Displacement (x) is the instantaneous distance of the moving object from its mean position.
• Amplitude (A) is the maximum displacement from the mean position.
• Period (T) is the time taken for one complete oscillation.
• Frequency (f) is the number of oscillations per unit time.
• Angular Frequency (ω) is the rate of change of angular displacement.
  • ω = 2πf
• Phase Difference (Φ) is the measure of how much one wave is out of step with another wave.
  • Φ = 2π ^t/_T
  • where T is time period, and t is the time lag between waves.

Simple Harmonic Motion

• Simple Harmonic Motion is acceleration proportional to the displacement and directed towards a fixed point.
• Requirements for S.H.M.:
  • A mass that oscillates
  • A position where the mass is equilibrium
  • A restoring force that acts to return mass to equilibrium; F ∝ -x
• Defining the equation of S.H.M.
  • a = -ω²x
    
• The negative sign in the equation shows that a and x are in opposite directions. a is always directed towards the mean position.

Equations of S.H.M.

• Displacement:
  • x = x_o sin ωt, OR x = x_o cos ωt
  • Either one of the above, depending on initial conditions
• Velocity:
  • v = ±ω√(x_o² – x²)
  • v = v_o cos ωt, OR v = -v_o sin ωt [x_oω = v_o]
  • Maximum velocity is at the equilibrium position and the minimum, 0, is at extremes.
• Acceleration:
  • a = -ω²x
  • a = -ω²(x_o sin ωt) OR a = -ω²(x_o cos ωt)

Graphs of S.H.M

Energy in S.H.M

• Kinetic Energy:
  • v = ±ω√(x_o² – x²)
  • E_k = ¹/₂ mv²
  • ∴ ¹/₂ mω²(x_o² – x²)
• Total Energy:
  • At x = 0, E_k is max and equal to total energy
  • ∴ E_total = ¹/₂ mω²(x_o² – 0²)
  • ∴ E_total = ¹/₂ mω²x_o²
• Potential Energy:
  • E_total = E_k + E_p
  • ∴ E_p = E_total – E_k
  • ∴ E_p = ¹/₂ mω²x_o² – [¹/₂ mω²(x_o² – x²)]
  • ∴ E_p = ¹/₂ mω²x²
• Graphs:
  

Examples

1. The needle of a sewing machine oscillates vertically through a total distance of 22 mm.
   
   The oscillations are simple harmonic with a frequency of 4.5 Hz. The cloth being sewn is positioned 8.0 mm below the needle when it is at its maximum height. Calculate, for the point of the needle:
   1. its maximum speed
      • Maximum speed can be calculated using v_o = x_oω
      • Firstly, we must find the angular velocity:
        • ω =  2πf = 2 × π × 4.5 = 28.3 rad s^-1
      • Next, we must find the amplitude. As the total vertical distance is 22 mm:
        • x_o = 22 ÷ 2 = 11 mm
      • Substitute the data into the first equation:
        • v_o = 28.3 × 11 × 10^-2 = 0.311 ms^-1
   2. its speed as it moves downwards through the cloth
      • To find the velocity at that point, use the equation: v = ω√(x_o² – x²)
      • We need to find the displacement for when the needle passes through the cloth.
        
        • 11 – 8 = 3 mm
      • Hence, substitute the values into the equation and calculate v:
        • v = 28.3 × √(11² – 3²) = 0.30 ms^-1

Damping

• Damping is the loss of energy and a reduction in the amplitude of an oscillating system caused by a force acting in the opposite direction of the system’s motion (e.g. friction).
• Light Damping: The system oscillates about an equilibrium position with a decreasing amplitude over a period of time.
  
• Critical Damping: The system does not oscillate, and it is the amount of damping required such that the system returns to its equilibrium position in the shortest possible time.
• Heavy Damping: The damping is so great that the displaced object never oscillates but returns to its equilibrium position very very slowly.
  

Practical Examples of Damping

Natural Frequency and Resonance

• Natural Frequency, f_o is the unforced frequency of oscillation of a freely oscillating object.
• Free Oscillation is oscillatory motion not subject to an external periodic driving force; it oscillates at its natural frequency.

• Forced Oscillation is oscillation caused by an external driving force; its frequency is determined by the driving force.
• Resonance is the maximum amplitude of vibration when an impressed frequency equals the natural frequency of vibration.

 

 

 

 

Damping and Resonance

• Effects of damping on frequency response of a system undergoing forced oscillations:
  • It decreases amplitude at all frequencies.
  • It slightly decreases the resonant frequency.
  • The resonant peak becomes flatter.
    

Purposes of Resonance

• Examples of Useful Purposes of Resonance:
  • Oscillation of a child’s swing.
  • Tuning of a radio receiver – the natural frequency of a radio is adjusted so that it responds resonantly to a specific broadcast frequency.
  • Using a microwave to cook food – It produces microwaves of frequencies equal to the natural frequency of water, causing the water molecules in the food to vibrate, thereby generating heat.
  • Magnetic Resonance Imaging (MRI) is used in hospitals to create images of human organs.

• Examples of the Destructive Nature of Resonance:
  • High-pitches sound waves can shatter fragile objects, e.g. shattering a glass when a soprano hits a high note.
  • Buildings that vibrate at natural frequencies close to the frequency of seismic waves collapse during earthquakes.
  • A car suspension system vibrates when going over bumps, which wold give large amplitude vibrations.