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I'm thinking about the mechanism by which one *starts* a playground
swing. I would like to think about it qualitatively, from both an
energy and a force standpoint.
If you just sit in a swing and move your legs back and forth at the
knees, not much happens. To start, one lays back and moves their legs
forward, more or less straightening them. The lay-back and legs-up
drops the center of mass. This drop in the potential energy is
accompanied by an increase in kinetic energy, hence the initial
motion.
But what are the forces for this initial motion? It would have to be
tension in the rope/chain, no? By laying back, the rope is bent at an
angle about the location where your hands are, and this changes the
tension in the rope, correct? This same bent requires by simple
geometry that the seat be higher than its original position - not a
problem as long as the CM has been lowered.