How to play Swing and Resonance
- Choose a test and read the lengths, springs, masses, and speeds on the card.
- Without friction the energy stays the same: it flows between motion, ½·m·v², and stored energy, m·g·h or ½·k·x².
- Pushes build a big motion only when they come once per natural period: resonance. Type your prediction and press Start.
- Watch the bars swap energy, or the graph grow when the pushes are in time.
Why do small pushes make a big swing?
An oscillator keeps trading energy. A swing let go from a height turns m·g·h into motion as it falls, ½·m·v² at the bottom, then climbs back to the same height; a block on a spring turns ½·k·x² into motion and back. Without friction the total never changes, so the speed at the bottom is √(2·g·h), and a spring pulled A stores ½·k·A², shared equally between motion and spring at x = A ÷ √2, not at A ÷ 2. Every oscillator also has its own natural period T₀. Push it once per period, always as it comes back, and each push adds to the last: the motion grows and grows. Push at the wrong rhythm and the pushes sometimes help and sometimes fight, so little builds up. This is resonance. It lets a child pump a swing high with small pushes, makes a car bounce on a road with evenly spaced bumps at one particular speed, and is why soldiers break step when they cross a bridge.
A focused learning model
There is no friction or air resistance, so energy only moves between motion and storage. The swing hangs on 2.45 m ropes, carries a 25 kg child, and stays at small angles; the block has a mass of 2 kg. Each push or bump is a short kick that adds the same speed: five of them, at the interval you give, within 16 s. The motion they make is added up exactly, and the run ends one natural period after the last one. Pushes and bumps play at triple speed; the other runs play in real time, for one full swing or 2 s. The swing is to scale; the spring and the car are not. Answers within 2% are right. g = 9.8 m/s².