Forced Oscillations & Resonance
Forced Oscillations & Resonance
- Push a swing at random moments and little happens; push in time with it and the amplitude grows and grows. That matching of frequencies has a name, a definition and a famous graph.
Driving an oscillator
- In a forced oscillation, a periodic driver makes the system oscillate at the driving frequency — not at its own natural frequency. The response amplitude depends on how close the two frequencies are.
- is the condition of maximum amplitude, reached when the driving frequency equals the natural frequency of the system. Those are the two marks, split exactly there (9702/41/M/J/24 Q4(a)) — the only resonance question of 2023–25, so expect it back.
- At resonance the system also absorbs energy from the driver at the greatest rate — that is why the amplitude builds.
- Asked to sketch the response curve, two features score: a single maximum at a non-zero frequency, placed at the natural frequency you measured earlier in the question (9702/41/M/J/24 Q4(c)). Note the curve does not start at zero amplitude at — a very slow driver still drags the system through its full stroke.
- More damping reshapes the curve three ways (in the figure): the peak gets lower, broader, and sits at a slightly lower frequency. Not examined in 2023–25, but it is syllabus wording — learn it with the curve.
Resonance for and against
- Useful resonance: microwave ovens drive water molecules at a natural vibration frequency; radio tuning circuits resonate at one station's frequency; instruments rely on resonating strings and air columns.
- Dangerous resonance: pedestrians walking in step drove London's Millennium Bridge into growing sway; earthquakes can drive buildings at their natural frequencies. The cure in both cases is engineered damping — absorb the energy faster than the driver supplies it.
Barton's pendulums
Hang several pendulums of different lengths from one string and set a heavy driver pendulum swinging: only the pendulum whose length (and so natural frequency) matches the driver builds a large amplitude. Every oscillator receives the same driving frequency — only the matching one resonates. It is the resonance curve acted out in hardware.