system
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human
Were does the inertia of a playground swing come from?

EDIT: Forget inertia, I meant momentum.

I just noticed I don't understand playground swing sets:

A kid sits on a playground swing set at rest. The system has zero net momentum. 

The child starts swinging by pumping its legs and throwing its body forward and backward. The system still has zero net momentum, as the child tries to shifts its center of gravity, the seat of the swing set accelerates forward and backward, keeping the center of gravity in place and total net momentum zero.

The child kept pumping its legs and due to its intuitive understanding of resonance, the net energy of the system increases (potential + kinetic, of course the child burns calories) with every swing and every pump. Once the swing really gets going, the child stops pumping and just sits in the seat.

As the child passes the lowest point of the pendulum motion, its momentum is obviously not zero. Were did that momentum come from? What balances it?

Do swing sets work in vacuum and with perfectly friction less bearings?

PS: The child never touched the ground with its legs. That would make it much easier.
gpt
Swings with children pumping their legs to increase their peak height have actually quite complicated physics that many people confuse with the simpler case of a pendulum with constant peak height and constant total energy. 

Where does the extra energy come from to increase the child's total energy (gravitational potential energy plus kinetic energy)? The energy ultimately comes from the child's muscles. How does energy get transferred from the muscles to the swing? The same way that muscles always create kinetic energy: by pushing off something or pulling towards something. If a child were floating in an empty vacuum with nothing to push, pull, or throw, she could not increase her linear kinetic energy, because of conservation of energy. So what is the child pushing off of or pulling against? The only thing that the child is connected to which is not moving with her is the bar to which the swing chains are attached on the top. Therefore, the child's pumping of her legs must effectively allow her to push off from or pull towards the bar. And this is indeed what happens. 

But it gets more complicated. The chains of the swing are typically connected to the bar so that force can only be transmitted between bar and chain in the radial direction and not in the tangential direction (this is, after all, what allows the swing to swing freely in the angular direction). Also, chains can only be pulled and not pushed. The only option to exert a force on the bar is therefore to pull the chain radially away from the bar. How can a child gain net momentum tangentially if she can only pull radially away from the bar?

The key is that a force can be redirected by an intervening agent (e.g. a ramp can redirect a downwards force into a sideways force). The intervening agent in this case is the child's body itself and how it is stuck by friction in the chair of the swing. When pumping, the child is effectively pulling radially away against the bar through the chains and diverting this radial force to a tangential force using her body and its connection to the swing seat and chains. Therefore, a force that is radially away from the bar becomes a tangential force that is able to angularly accelerate the child and increase her peak height. 

If the seat of the swing were large, flat and frictionless, and the child was careful not to hold or touch the chains, then there would be no way to divert the radial pull on the chains to a forwards push. The child would not able to increase her peak height. She would just slide around on the chair (or fall off) when she pumped her legs, but would not get it swinging.

> Do swing sets work in vacuum and with perfectly friction less bearings?

Yes. The child is not pushing off the air and is not directly pushing tangentially off the bar through the chains. She is pulling radially away from the bar.