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I feel this might be a FAQ but I would love a definitive answer.

Imagine a frictionless stationary idealised child's playground swing. If you are sitting on the seat of the swing, is it possible in principle to set it into motion by simply moving your body and pulling on the chains of the swing? If so, is there a simple explanation for how you can change your center of mass without an external force acting on you?

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    $\begingroup$ Yes, if you spit you'll start to move. $\endgroup$ – jinawee Jun 5 '14 at 19:06
  • $\begingroup$ @jinawee Good point. I didn't have that sort of thing in mind. I have clarified the question a little. $\endgroup$ – felix Jun 5 '14 at 19:07
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    $\begingroup$ There is an external force. The swing is connected to you, and you exert force on the swing. The swing is connected to the swing set, and the swing is exerting force on the swing set. And gravity is acting on both you and the swing. $\endgroup$ – Peter Shor Jun 5 '14 at 19:16
  • $\begingroup$ @PeterShor This is very interesting thank you. Just to finally clarify, the swing chains are only attached to the swing set at their tops. Is it right that you don't require any friction between the chains and the swing set at all? I am trying to imagine the direction of the forces between the swing set and the swing chains. $\endgroup$ – felix Jun 5 '14 at 20:33
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    $\begingroup$ I think you have to assume that the swing chains are short enough that they can exert a non-trivial horizontal force on the top of the swing set when you lean back and pull the chains back, or do something similar. It's a small force (which is why it takes a while to start swinging from a stationary swing). $\endgroup$ – Peter Shor Jun 5 '14 at 20:38
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Yes. A swing does not depend on friction. The way you start up is by leaning back and pulling hard with your hands placed on the ropes. Your hand-hold positions effectively create a compound pendulum, so your mass is offset from null. By timing your shift in weight back to a "sitting-up" position, you essentially add energy to the pendulum system in phase and frequency with the full-length pendulum's oscillation.

I'm sure there are some pictures of this somewhere :-)

EDIT: I should have said, as Peter Shor did, that this makes use of gravitational force.

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  • $\begingroup$ Thank you. In practice you always take a running jump to start a swing swinging. It is interesting if this is not in fact necessary. $\endgroup$ – felix Jun 5 '14 at 19:34
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    $\begingroup$ As an "experimental verification," I've done it before, starting a swing from complete rest by just moving my body. $\endgroup$ – NeutronStar Jun 5 '14 at 19:51
  • $\begingroup$ That is interesting and surprising. I had always assumed that if this was possible it was because of friction between where the swing chains meet the static part of swing. $\endgroup$ – felix Jun 5 '14 at 20:37
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    $\begingroup$ How would the center of mass of rope + board + body ever get away of the vertical through the point of rotation if there is no friction? How would you ever get moving on a perfectly slippery horizontal plane? $\endgroup$ – coproc Apr 3 '15 at 12:27
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    $\begingroup$ sure, a frictionless plane is different; but what makes the crucial difference? whatever you do, the center of gravity of (rope + board + body) will at most move up and down. If the body leans back then the board + rope will move forward and the center of gravity of (rope + board + body) still remains on the same vertical line beneath the rotation point of the rope. I dont see how without touching the ground you will ever get the cog. moving back or force - like one will never get a boat moving in one dircetion by only running back and force inside (when neglecting friction): the cog is fixed. $\endgroup$ – coproc Apr 3 '15 at 20:03
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As Peter Shor writes, there is external force. Imagine the swing was on rollers: it would swing back and forth as the child swings forth and back. And the horizontal position of the mass center would not move.

The process is conversion of muscular energy into kinetic energy, then via the swing attachment, to potential energy. And loop: produce kinetic energy again when it reaches 0 (and potential energy is maximum).

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This phenomenon is independent of any force other than gravity and those of the system child and the swing. It is simply due to conservation of momentum and what is known as parametric resonance. When the child at rest moves his legs, conservation of momentum implies that his body moves in the opposite direction and appropriate timing of this motion adds energy to the system by leveraging gravity. The cause of this is independent of whether the hinge is fixed or not in the horizontal direction. The inertia of the chain is enough (the chain will not form a straight line, therefore the force on the hinge will be closer to the vertical than the line connecting the hinge and the swing. This is a small effect and I don't know if it can be leveraged in the real world.) to delay the motion of the hinge enough so that a restoration torque due to gravity is formed.

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