Energy in Simple Harmonic Motion, Damping and Resonance
An oscillator trades its whole energy between a kinetic store and a potential one twice per cycle, and the total stays fixed only while nothing dissipates. Damping drains it, and driving at the natural frequency pumps the amplitude up until the losses catch the input.
What a learner can do afterwards
- Sketches kinetic, potential and total energy against displacement for one oscillation and explains the shape of each
- Distinguishes light, critical and heavy damping by what happens to a displaced system, and gives a use for each
- Explains why amplitude peaks near the natural frequency and what heavier damping does to the height and width of that peak
1 · Read
In an ideal oscillator, energy flows back and forth while the total stays fixed. At the extremes the mass pauses, so all energy sits in the spring, while at the centre the spring is relaxed and all energy is motion. In between, the two stores share the total, with more in the spring further out. Plotted against displacement, kinetic energy is an upside down parabola peaking at the centre, potential energy is an upright parabola peaking at the edges, and total energy is a flat line.
Real oscillators lose a little energy each cycle, and this damping picks one of three stories. Light damping swings many times with slowly shrinking amplitude, like a tuning fork. Critical damping returns to rest in the shortest time without overshooting, which car suspensions aim for. Heavy damping creeps slowly back with no oscillation at all, like a door closer that must never slam.
Pushing once per cycle feeds energy in, and the response peaks when you push at the natural frequency. That peak is resonance, familiar from pushing a swing. It sits near the natural frequency and grows taller and narrower when damping is light. Heavier damping lowers the peak and widens it, spreading a smaller response over more frequencies. Radio tuners use sharp resonance to select one station, while shock absorbers use strong damping to stop resonance ever building up.
Read an observed return to rest to name the regime: overshoot and shrink means light, fastest clean settle means critical, slow creep means heavy. Then match the use to the regime.
Energy trades stores each cycle, damping picks the return story, and resonance peaks at the natural rhythm.
The rewritten readings were written for this dataset.
More to read
Energy during a spring oscillation
An ideal horizontal mass-and-spring oscillator keeps a constant total energy. Energy transfers between the mass's motion and the spring's deformation. At either turning point, the mass stops briefly and its energy is entirely elastic potential energy. At equilibrium, the spring is unstretched and the energy is entirely kinetic. Between these positions, both forms are present. Kinetic energy is proportional to speed squared.
Elastic potential energy increases with displacement squared. Its displacement graph is an upward-opening parabola. Kinetic energy is the constant total minus that potential energy, so its graph opens downward and reaches its maximum at equilibrium. Across the motion's range, total energy appears as a horizontal line.
Three ways motion dies away
Friction and air resistance remove energy from real oscillators. With light damping, repeated swings gradually become smaller. A tuning fork or an unpushed playground swing can behave this way.
Critical damping gives the fastest return to equilibrium without oscillation. Vehicle suspensions balance a quick return against other ride requirements rather than always using exactly critical damping. With heavier, overdamping, the return is slower and still has no oscillation. A slowly closing door can show this behaviour. Observing whether motion oscillates, and how quickly it settles, helps identify the damping regime and its possible uses.
Driving an oscillator into resonance
Repeated, correctly timed pushes transfer energy to an oscillator. With light damping, its largest displacement response occurs near its natural frequency. This is resonance, as when pushing a swing or breaking a glass with a matching note.
Light damping produces a high, narrow resonance peak. Increasing damping lowers and broadens it; sufficiently strong damping can remove the displacement peak. Radio tuning uses a narrow resonance to distinguish stations. Shock absorbers dissipate energy, reducing the build-up of large resonant motion.
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2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.