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How can we swing faster and faster without touching the ground? I mean, there's no external force on the swing + human system, right? Or is it because of gravity?

EDIT : Another possibility : are we using the energy stored in our body (as food and everything) to increase the speed?

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Even if we use energy stored in our bodies, we cannot produce a net momentum in a system that has only internal forces on it. So you know the swing must have an external force on it. It can't be gravity by itself, because gravity only points downward, yet at the bottom of its motion, a swing clearly has a horizontal momentum. So we must look for something external to the swing that is touching the swing. For that, we must look at the attachment of the swing at its top. Inspect that attachment closely, and you will see that it is capable of producing external horizontal forces, as required. The art of learning to swing is the art of learning how to move your body (and use that stored energy) to get the attachment to produce horizontal forces in synch with the pendulum frequency of the swing.

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Yes , what you think in the Edit part of your question is correct.

You apply your own effort in swinging as you apply torque to swing faster. This is both to speed up and also to counteract the resistance offered by air .

The torque acts .Your swinging action along with gravity helps to increase the angular velocity of the system.This uses up your internal energy. That's why you feel exhausted after swinging a lot.

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As you move the position of your body your center of mass changes. Gravity, pulling down, naturally shifts your center of mass to the lowest point of the arc, which by pulling along the vector of the chain induces a horizontal force. At the opposing apex you shift your center of mass again, to the opposite extreme, inducing more horizontal force.

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