PLAYGROUND SWING ================ An independent reimplementation. Written from first principles for this app; no code, art, data or text was taken from any other program. WHAT IS BEING CREDITED ---------------------- A playground swing has no author, no year and no publisher. It is a rope and a plank, older than the physics that explains it. What this app owes its content to is a body of published work on how a swing is pumped, and to one reference table of special functions. Curry, S. M. "How children swing." American Journal of Physics 44, 924 (1976). doi:10.1119/1.10230. Parametric-amplifier framing; energy grows approximately exponentially under constant pumping, at a rate independent of the child's mass. READ: abstract only. The full text is paywalled. Tea, P. L. and Falk, H. "Pumping on a swing." American Journal of Physics 36, 1165 (1968). doi:10.1119/1.1974385. The discrete-jump standing model. READ: the publisher's rasterised first-page preview only. Burns, J. A. "More on pumping a swing." American Journal of Physics 38, 920 (1970). doi:10.1119/1.1976498. The continuous form th'' + (1/l)[2 l' th' + g th] = 0, which this engine reduces to exactly when the rider moves only along the rope. READ: first page only. Case, W. B. and Swanson, M. A. "The pumping of a swing from the seated position." American Journal of Physics 58, 463 (1990). doi:10.1119/1.16477. Seated pumping as a driven oscillator plus parametric terms, the driving traced to conservation of angular momentum about the support. READ: abstract only. Case, W. B. "The pumping of a swing from the standing position." American Journal of Physics 64, 215 (1996). doi:10.1119/1.18209. Finds driving terms dominant even in the standing posture at ordinary playground amplitudes, explicitly contradicting the earlier parametric framing. READ: abstract only. Post, A. A., de Groot, G., Daffertshofer, A. and Beek, P. J. "Pumping a playground swing." Motor Control 11(2), 136 (2007). doi:10.1123/mcj.11.2.136. Eighteen people motion-captured pumping from standstill: driven oscillation major, parametric subordinate, parametric share rising with amplitude. READ: abstract only; the body is restricted. Wirkus, S., Rand, R. and Ruina, A. "How to pump a swing." College Mathematics Journal 29(4), 266 (1998). doi:10.1080/07468342.1998.11973953. NOT READ: paywalled, and this journal publishes no abstract for that era. Listed because it is the standard citation, not because anything here came from it. NIST Digital Library of Mathematical Functions (DLMF), section 28.6. https://dlmf.nist.gov/28.6 The Mathieu transition-curve series a1, b1, a2, b2, a3, quoted verbatim in js/mathieu.js and checked there against an independent Floquet computation. U.S. Consumer Product Safety Commission, Public Playground Safety Handbook, publication #325. https://www.cpsc.gov/s3fs-public/325.pdf Pivot height 47-96 in; preschool belt seat 14-28 in. This app's beam sits at 94.5 in and its seat at 15.7 in. ASTM F1487 itself is commercial and was not read. WHAT DIFFERS FROM A REAL SWING ------------------------------ 1. No air resistance and no pivot friction. A real swing loses a few percent per cycle. 2. The rope is inextensible and the motion is strictly planar. No chain twist, no stretch. 3. The rider is two numbers -- a distance along the rope and an angle about their own centre of mass -- moved by a critically damped servo standing in for muscle. 4. The round ends the moment the chain tension would go negative, at 90 degrees of amplitude. A real swing jolts and the chain snaps taut; that is not modelled. 5. Jumping off is a bare projectile: no drag, no landing, no roll. 6. Gravity is exactly 9.80665 m/s^2, the 1901 CGPM standard value, not a local one. 7. The rider cannot let go of the chains, stand on one foot, or twist. WHAT IS ORIGINAL HERE --------------------- - The equation of motion, derived from the Lagrangian for a rigid rider with a rheonomic body coordinate, and its exact reduction to Burns' equation as a special case. - The exact result that a sinusoidal squat at the swing's own frequency transfers no energy whatever, at any depth and any phase, because g/l and l''/l cancel identically. - The closed form D = m g Ic L alpha0 / M0 for the seated driving torque, showing it is exactly proportional to the rider's own moment of inertia and vanishes for a point mass. - The per-half-swing law (l_long / l_short)^3 for the impulsive pump, checked three ways. This disagrees with Tea and Falk (1968) equation 4 as transcribed from a page image; the discrepancy is reported, not resolved, because the full text could not be read. - The renderer, the game, the harnesses and every figure on the page. NO NETWORK, NO ACCOUNTS, NO COST -------------------------------- This app makes no network requests of any kind after it loads. It has no backend, no model call and no analytics. Best scores live in this browser under the key "playground-swing:best" and nowhere else.