Free oscillation and natural frequency
An object that is displaced and released oscillates on its own at a fixed natural frequency (f₀). This is called free oscillation. In an ideal case with no energy loss, the amplitude would stay constant forever.
Damping
In reality, an oscillating system loses energy to its surroundings through friction and air resistance. As energy is lost, the amplitude decreases with time — this is damping. Importantly, damping reduces the amplitude but does not change the natural frequency.
- External damping: energy lost to the surrounding medium (air, water).
- Internal damping: energy lost within the material of the oscillating body itself.
Forced oscillation and resonance
If a periodic external force drives a system, the system undergoes forced oscillation at the driving frequency. Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance, energy is transferred most efficiently, so the amplitude becomes maximum.
Key formula / Key formula
Resonance condition: f_driving = f₀
For a simple pendulum, natural frequency depends on length: a longer pendulum has a smaller f₀. Using T = 1/f, a period of 2 s gives f₀ = 0.5 Hz.
Worked example
A pendulum has a natural period of 2.0 s. At what driving frequency will it resonate?
Natural frequency: f₀ = 1/T = 1/2.0 = 0.5 Hz.
Resonance occurs when the driving frequency equals 0.5 Hz, giving the largest swing.
Everyday examples
Resonance is useful (tuning a radio, a microwave oven heating water molecules) but can be destructive (a bridge or building oscillating dangerously if wind or an earthquake drives it near its natural frequency). Barton's pendulums show that only the pendulum whose length matches the driver oscillates with the largest amplitude.
Remember
- Damping lowers amplitude, not natural frequency.
- Resonance: driving frequency = natural frequency → maximum amplitude.
- Greater damping gives a smaller resonance peak.