Resonance & Energy in SHM
Watch kinetic and potential energy trade places as a mass oscillates, and see why matching the driving frequency triggers resonance. SHM energy and resonance, made visual.
Push a child on a swing at just the right moments and a series of gentle nudges builds into a soaring arc. Push at the wrong rhythm and you fight the swing to a standstill. The difference is resonance— and it begins with how a simple oscillator stores and trades its energy.
Energy on a seesaw
In simple harmonic motion the total mechanical energy stays constant, but it never sits still: it swaps continuously between kinetic energy (KE) and potentialenergy (PE). At the extremes the mass is momentarily still — all its energy is potential. Racing through the middle it moves fastest — all kinetic. Everywhere in between, the total E = KE + PE is the same fixed amount.
Think of a skateboarder in a half-pipe: highest and slowest at the top of each side, lowest and fastest at the bottom, endlessly trading height for speed.
At what displacement does an oscillator carry equal amounts of kinetic and potential energy?
The potential energy grows as ½kx², while the total energy is ½kA². Setting PE to half of the total gives ½kx² = ½ · (½kA²), so x² = A²/2 and x = A/√2 ≈ 0.71A. Only past 71% of the way out does potential energy finally overtake kinetic.
Trading kinetic for potential
Set the mass on the spring oscillating below and watch the three graphs. Notice how the velocity peaks exactly as the mass crosses the middle (maximum KE) and drops to zero at the extremes (maximum PE), while the acceleration always points back towards equilibrium.
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A mass on a spring in simple harmonic motion, with graphs of displacement, velocity and acceleration. At the extremes the mass is momentarily still with all energy potential; at the equilibrium position it moves fastest with all energy kinetic. Acceleration always points back towards equilibrium, a = −ω²x.
Driving at the natural frequency
Left alone, any oscillator vibrates at its own natural frequency, f₀. Now drive it with a repeated external push. When the driving frequency matches f₀, each push arrives exactly in step with the motion, adding a little energy every cycle. The amplitude climbs to a large maximum — this is resonance.
The swing is the classic case: time your pushes to its natural rhythm and the arc grows and grows. Drive it far above or below f₀ and your pushes fall out of step, sometimes helping and sometimes fighting the motion, so the amplitude stays small.
Damping and the resonance peak
Real systems lose energy to friction and air resistance — this is damping. Damping steadily removes energy, shrinking the amplitude of a free oscillation over time. When a system is driven, damping limits how high the resonance peak can rise.
Common mistakes
Practice
Why does pushing a swing at the wrong rhythm fail to build up a big amplitude?
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Because energy is only added when each push is in stepwith the swing’s natural frequency. At the wrong rhythm the pushes arrive out of step — sometimes with the motion, sometimes against it — so on average they add little or no energy, and the amplitude stays small.
Resonance in the wild
The same effect tunes a radio to a single station, lets a singer shatter a wine glass, and has forced engineers to redesign footbridges that swayed in step with pedestrians. Wherever a driving frequency meets a natural one, amplitudes can grow dramatically.
Frequently asked questions
How does energy change during simple harmonic motion?+
What is resonance?+
What is the natural frequency?+
What does damping do?+
The ScholarsGate Physics Team
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Written and reviewed by ScholarsGate tutors who teach A-Level and undergraduate physics. Every explainer is checked against the AQA, Edexcel, OCR and CIE specifications.
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