The Nature of Resonance (HL IB Physics)

Free Oscillations

• Free oscillations occur when there is no transfer of energy to or from the surroundings
  • This happens when an oscillating system is displaced and then left to oscillate

• In practice, this only happens in a vacuum. However, anything vibrating in air is still considered a free vibration as long as there are no external forces acting upon it
• Therefore, a free oscillation is defined as:

An oscillation where there are only internal forces (and no external forces) acting and there is no energy input

• A free vibration always oscillates at its resonant frequency

Forced Oscillations

• In order to sustain oscillations in a simple harmonic system, a periodic force must be applied to replace the energy lost in damping
  • This periodic force does work on resistive forces (i.e. the force that decreases the amplitude of the oscillations), such as air resistance
  • It is sometimes known as an external driving force

• This period force creates forced oscillations (or vibrations) is are defined as:

Oscillations which are produced by a periodic external force

• Forced oscillations are made to oscillate at the same frequency as the external oscillator creating the external, periodic driving force
  • This means the driving force can change the frequency of the oscillator

Resonance

• The frequency of the forced oscillations on a system is referred to as the driving frequency f
• All oscillating systems have a natural frequency f₀ which is defined as

The frequency of an oscillation when the oscillating system is allowed to oscillate freely

• Oscillating systems can exhibit a property known as resonance when

driving frequency f  =  natural frequency f₀

• When the driving frequency approaches the natural frequency of an oscillator, the system gains more energy from the driving force
  • Eventually, when they are equal, the oscillator vibrates with its maximum amplitude
  • This is resonance

• Resonance is defined as:

When the frequency of the applied force to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations is at its maximum

Example of resonance: a child pushed on a swing

• Every system (in this case, the swing and the child) has a fixed natural frequency
• A small push (the driving force) after each cycle increases the amplitude of the oscillations, resulting in the swing's motion back and forth
• The frequency at which the swing is pushed is the driving frequency
• When the driving frequency is equal to the natural frequency of the swing, resonance occurs
• If the driving frequency is slightly lower or higher than the natural frequency, the amplitude will increase but to a lesser extent than if they were equal

• This is because, at resonance, energy is transferred from the driver to the oscillating system most efficiently
  • Therefore, at resonance, the driving force transfers the maximum kinetic energy to the system
  • In this case, the child will swing the highest when resonance occurs

Pushing a child on a swing is an example of how forced oscillations can produce resonance