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Oscillations - Simple Harmonic Motion

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 = xo sin ωt, OR x = xo cos ωt
    • Either one of the above, depending on initial conditions
  • Velocity:
    • v = ±ω√(xo2 – x2)
    • v = vo cos ωt, OR v = -vo sin ωt [xoω = vo]
    • Maximum velocity is at the equilibrium position and the minimum, 0, is at extremes.
  • Acceleration:
    • a = -ω2x
    • a = -ω2(xo sin ωt) OR a = -ω2(xo cos ωt)
Graphs of S.H.M

Energy in S.H.M

  • Kinetic Energy:
    • v = ±ω√(xo2 – x2)
    • Ek = 1/2 mv2
    • 1/22(xo2 – x2)
  • Total Energy:
    • At x = 0, Ek is max and equal to total energy
    • ∴ Etotal1/22(xo2 – 02)
    • ∴ Etotal1/22xo2
  • Potential Energy:
    • Etotal = Ek + Ep
    • ∴ Ep = Etotal – Ek
    • ∴ Ep = 1/22xo2 – [1/22(xo2 – x2)]
    • ∴ Ep 1/22x2
  • 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 vo = xoω
      • 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:
        • xo = 22 ÷ 2 = 11 mm
      • Substitute the data into the first equation:
        • vo = 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√(xo2 – 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, fo 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.
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