Resonance (AQA A Level Physics): Revision Note

Resonance

• The frequency of forced oscillations is referred to as the driving frequency ($f$) or the frequency of the applied force

• All oscillating systems have a natural frequency ($f_0$), which is the frequency of an oscillation when the oscillating system is allowed to oscillate freely

• Oscillating systems can exhibit a property known as resonance

• 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 increases significantly

• For example, when a child is pushed on a swing:

  • the swing plus the child has a fixed natural frequency

  • a small push after each cycle increases the amplitude of the oscillations to swing the child higher

    • The frequency at which this push happens is the driving frequency

  • when the driving frequency is exactly equal to the natural frequency of the swing oscillations, resonance occurs

  • if the driving frequency does not quite match the natural frequency, the amplitude will increase but not to the same extent as when resonance is achieved

• This is because, at resonance, energy is transferred from the driver to the oscillating system most efficiently

  • Therefore, at resonance, the system will be transferring the maximum kinetic energy possible

• A graph of driving frequency $f$ against amplitude A of oscillations is called a resonance curve

• It has the following key features:

  • when $f<f_0$, the amplitude of oscillations increases

  • at the peak where $f=f_0$, the amplitude is at its maximum — this is resonance

  • when $f>f_0$, the amplitude of oscillations starts to decrease

The effects of damping on resonance

• Damping reduces the amplitude of resonance vibrations

• The height and shape of the resonance curve will therefore change slightly depending on the degree of damping

  • Note: the natural frequency $f_0$ of the oscillator will remain the same

• As the degree of damping is increased, the resonance graph is altered in the following ways:

  • The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers

  • The resonance peak broadens

  • The resonance peak moves slightly to the left of the natural frequency when heavily damped

• Therefore, damping reduces the sharpness of resonance and reduces the amplitude at resonant frequency

Resonance effects

• Resonance occurs for any forced oscillation where the frequency of the driving force is equal to the natural frequency of the oscillator

• Examples include:

  • an organ pipe, where air resonates down an air column setting up a stationary wave in the pipe

  • glass smashing from a high pitched sound wave at the right frequency

  • a radio tuned so that the electric circuit resonates at the same frequency as the specific broadcast

• A mechanical system commonly used to show resonance is Barton's pendulums

• A set of light pendulums labelled A-E are suspended from a string

  • A heavy pendulum X, with a length L, is attached to the string at one end and will act as the driving pendulum

• When pendulum X is released, it pushes the string and begins to drive the other pendulums

• Most of the pendulums swing with a low amplitude but pendulum C with the same length L has the largest amplitude

  • This is because its natural frequency is equal to the frequency of pendulum X (the driving frequency)

• The phase of the oscillations relative to the driver are:

  • pendulums E and D with lengths < L are in phase

  • pendulum C with length = L is 0.5π out of phase

  • pendulums B and A with lengths > L are π out of phase