# Is there an optimum length for a swing for a childs mass?

If you want to maximize the maximum velocity a child could go, what would be the optimum height?

If you wanted to maximize the efficiency of a child "pumping" their legs to gain velocity, what would be the optimum height?

I hope that I am correctly understanding how a child pumping their legs makes it possible for them to gain momentum. They shift the center of mass of the swing+child outside the normal position of being in-line with the rope/chain. By shifting the center of mass behind t's normal position, they are allowing gravity to place a small amount of force in their direction of motion. When their center of mass and the rope is parallel with the force of gravity (like a plumb bob), they cannot impart any additional velocity on their swing motion. When they are at the apex, they can impart the most force.

I imagine too short a swing, and the child will not have a very high maximum velocity, and the time between pumps will be too short.

I imagine too long, and the wind resistance will be too great, and the childs mass compared to the entire swing will be too little.

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" When their center of mass and the rope is parallel with the force of gravity (like a plumb bob), they cannot impart any additional velocity on their swing motion." Not so I think. At the bottom of the swing they can raise their centre of mass by pulling on the chain and changing body position thus reducing their moment of inertia about the point of suspension. To conserve angular momentum the angular speed and hence the linear speed has to increase. Then sit down again at the top of the swing. – Farcher Mar 16 at 17:03

Well, I suspect that $M_{swing} \gg M_{legs}$ is pretty bad for the efficiency in that meaning. With a child's legs being less than 10 kg, that suggests that swing massing over ~100 kg would be impossible to move much. Of course 100 kg of steel cable makes one heck of a swing. I suspect that air resistance limits the height before we get that long, which makes my answer incomplete. In my defense I assumed you were talking about reasonably realizable swings. The tallest one I've used was about 6 meters, and I was able to get it near horizontal. That was fast. – dmckee Feb 6 '12 at 19:00