All subjects · Physics · WPH15 (Unit 5)

Free and forced oscillations, resonance and damping

Pushing a swing at the right moment makes it go higher and higher. This is resonance, and damping controls how dramatic it is.

Free and forced oscillations

Resonance

When the driving frequency equals (or is close to) the natural frequency, energy is transferred most efficiently and the amplitude becomes very large: resonance. A graph of amplitude against driving frequency shows a peak at about f_0. Examples: pushing a swing, musical instruments, microwave ovens and MRI scanners (useful); bridges and buildings shaking (harmful).

Damping

Damping removes energy from an oscillation, by friction, air resistance or deliberately (for example oil dampers). Increasing damping:

Light damping: many oscillations that slowly die away. Heavy damping: slow return to equilibrium with no oscillation.

Plastic deformation

Ductile materials, such as some metals, can be used to absorb energy. When they deform plastically, work is done that is not returned as the material does not spring back. This energy is dissipated, which reduces the amplitude of oscillations. Buildings in earthquake zones and some car parts use this idea.

Worked example

A mass of 0.40 kg on a spring of stiffness 160 N/m is driven by a vibrator. At what frequency will it resonate?

  1. f_0 = (1/2π) sqrt(k / m) = (1/2π) sqrt(160 / 0.40)
  2. = 20 / 2π

Answer: f_0 = 3.2 Hz

Key idea: Free: natural frequency. Forced: driving frequency. Resonance: maximum amplitude when driving frequency ≈ natural frequency. Damping lowers and broadens the peak.

Check yourself

  1. A 0.40 kg mass on a spring of stiffness 160 N/m is driven by a vibrator. Find the driving frequency for resonance, in Hz, to 3 s.f.
    Show answer

    3.18 Hz. f_0 = (1/2π) sqrt(160 / 0.40) = 20 / 2π = 3.18 Hz.

  2. A simple pendulum is 0.80 m long. What driving period will make it resonate, in s, to 3 s.f.?
    Show answer

    1.79 s. Resonance when T_drive = T_0 = 2π sqrt(0.80 / 9.81) = 1.79 s.

  3. What is a free oscillation?
    • One driven by a periodic force
    • One with zero amplitude
    • One where the object oscillates at its natural frequency with no driving force
    • One that never loses energy
    Show answer

    One where the object oscillates at its natural frequency with no driving force. It is displaced, released and left alone.

  4. At what frequency does a forced oscillator vibrate?
    • Its natural frequency
    • The driving frequency
    • Twice the driving frequency
    • Zero
    Show answer

    The driving frequency. The driver sets the frequency; the amplitude depends on how close it is to f_0.

  5. What is resonance?
    • The decrease of amplitude due to friction
    • Oscillation at twice the natural frequency
    • Oscillation with no energy transfer
    • A large increase in amplitude when the driving frequency is close to the natural frequency
    Show answer

    A large increase in amplitude when the driving frequency is close to the natural frequency. At resonance the driver transfers energy most efficiently.

  6. How does increasing the damping change a resonance curve?
    • The peak becomes higher and narrower
    • The peak becomes lower and broader
    • The peak moves to a much higher frequency
    • The curve becomes a straight line
    Show answer

    The peak becomes lower and broader. More energy is removed each cycle, so the amplitude cannot build as much.

  7. How does plastic deformation of a ductile material reduce the amplitude of oscillations?
    • It makes the material stiffer, increasing the natural frequency
    • It stores the energy elastically
    • It increases the driving frequency
    • Energy is used to deform the material permanently and is not returned
    Show answer

    Energy is used to deform the material permanently and is not returned. Plastic deformation dissipates energy rather than storing it.

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.