Oscillations: simple harmonic motion and resonance
A pendulum or a mass on a spring swings back and forth with its own steady rhythm, whatever the size of the swing. Energy flows between motion and storage, and pushes in time with the rhythm build a big motion: T = 2π·√(L ÷ g), T = 2π·√(m ÷ k), E = ½·k·A².
The period is the time for one beat; the frequency is how many beats fit in a second. Turn tempos into periods and hertz into beats per minute, count beats, and find when two metronomes click together.
Amplitude is how far an oscillation swings from the middle, not from top to bottom. Read it off a harbour post, see the period stay the same on a bigger swell, and find distances, speeds, and energies.
A seismograph turns shaking into a wavy line. Read the period from the peak spacing and the paper speed, the frequency from the period, and the ground's tiny amplitude from the magnified trace.
A pendulum's period depends only on its length and on g: T = 2π·√(L ÷ g). Build a seconds pendulum, time a short one, double a period, swing one on the Moon, and try a heavier bob.
A swinging ball trades height for speed and back: v = √(2·g·h). Find speeds at the bottom and partway down, the rope's pull, Galileo's peg, and the angle a given speed reaches.
T = 2π·√(L ÷ g) is only exact for tiny swings, and real swings fade. Find a big swing's period, a clock's error, how a swing shrinks, its half-life, and the energy it loses.
A spring pulls back with k·x, and a mass on it bounces with T = 2π·√(m ÷ k). Find a stretch, a period, a spring, a top speed, and a period from the stretch alone.
Springs side by side share the load and act stiffer; end to end, each carries it all and they act softer. Predict stretches, the stiffness of half a spring and of a mixed pair, and a bounce.
A squashed spring stores ½·k·x² and hands it on when let go. Launch a pinball and a rocket, aim for a platform, see what doubling the push does, and stop a cart with a spring.
An oscillator trades motion for stored energy and back, and small pushes in time with it build a big swing. Time a swing's pushes, find speeds and energies, and the bumps that make a car resonate.