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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, 2014 at 19:06
  • $\begingroup$ @jinawee Good point. I didn't have that sort of thing in mind. I have clarified the question a little. $\endgroup$
    – Simd
    Jun 5, 2014 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. $\endgroup$ Jun 5, 2014 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. $\endgroup$
    – Simd
    Jun 5, 2014 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). $\endgroup$ Jun 5, 2014 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. $\endgroup$
    – Simd
    Jun 5, 2014 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. $\endgroup$ Jun 5, 2014 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. $\endgroup$
    – Simd
    Jun 5, 2014 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, 2015 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, 2015 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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Starting a Swing from Rest

BRYAN F. GORE
Central Washington State College
Ellensburg, Washington 98926
(Received 3 August 1970)

When the upright occupant of a swing at rest falls backward and catches himself in a reclining position, the swing seat moves forward both during and after the occupant's rotation. This is in apparent contradiction to the backward motion of the occupant's center of mass (c.m.) expected during his rotation.¹ Nevertheless, the unmarked c.m. does move backwards. Its motion is simply masked by the forward motion of the seat due to the occupant's rotation. This may be easily seen if the position of the seat is marked at the end of the rotation and compared with its equilibrium position after friction has stopped the motion.

While efficient pumping of a swing is done so that the tangential force (associated with the torque changing the occupant's rotation about his c.m.) is always parallel to the swing's motion, the first pump takes place during a time shorter than a quarter of the swing's period. Therefore, the swing's motion does not reverse between imposition of the torques starting and stopping the occupant's rotation, and the tangential force due to the stopping torque opposes the motion imparted by the starting torque. It is straightforward to show that after the first pump the center of mass of the occupant, then displaced from its equilibrium position, will be moving slowly toward it.

The tangential impulse imparted to the c.m. during the pump has two components, one due to gravity and the other imparted by the torques. Since the tangential component of the rope's tension is proportional to the torque about the c.m., its contribution to the tangential impulse is proportional to the net angular impulse about the c.m. (assuming a constant moment of inertia). At the conclusion of the pump, rotation is halted, so the net angular impulse is zero. Consequently, the total tangential impulse is just that contributed by gravity. A slow motion toward the equilibrium position is thus predicted, which is just what is observed.

¹ B. F. Gore, Amer. J. Phys. 38, 378 (1970).

Transcribed and edited (by adding a missing closing parenthesis, splitting a long paragraph into two, and adding an initialism hint after center of mass) by me.
Source: https://doi.org/10.1119/1.1986146

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Moving your weight back is equivalent to moving the pivot forward, like having a ball on a string and moving your hand forward, then, when it moves forward, moving your hand back, but opposite, like the ball leaning back instead of your hand moving forward. I have since worked out you can also propel it forward by moving your legs forward instead of back, here "How does moving your legs forward on a swing propel it forward instead of backward?"

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