The pumping of a swing is almost a magical process. There is no external agent driving it higher; the driving is entirely internal. It seems to defy the laws of physics. Of course it is also magical as an experience as pointed out by Robert Louis Stephenson in his Child's Garden of Verses:
How do you like to go up in a swing,
Up in the air so blue?
Oh, I do think it the pleasantest thing
Ever a child can do!
At an earlier time it was understood as a parametric oscillator. If one has a ball on a string and raises it at its lowest point the swinging motion will increase as shown in the video clip.
Applying this to the playground swing the swinger must raise his or her body as the swing passes through the lowest point and lower themselves near the extremes of the motion. This is shown in the following video clip:
Yes yes, very nice, but that is not the way most people do it. Here is the way the other kids at the playground pump a swing:
The swinger is rather quiet as he passes through the lowest point and rocks forward at the most forward point and backward at the most rear point of the motion. For the seated swinger the situation is similar:
Again the swinger rotates either forward or backward near the ends of the swinging motion.
The next video clip shows a "swinger" made up of a wheel attached to the lowest point of a pendulum. Strings are attached allowing one to rotate the wheel back and forth. When the wheel is rotated at the ends of the motion the motion of the swing increases rapidly.
When the physics of this system is analyzed one finds that it follows a different equation. This motion is described as a driven oscillator. Although this model would more accurately represent a seated swinger, the same mechanism occurs for the standing swinger. Both the parametric oscillator and driven oscillator mechanisms are capable of pumping the swing, but the driven oscillator is more effective for the amplitudes at which most people swing. The two mechanisms also differ significantly in the equations they follow and the motion of the swinger.
To understand these two mechanisms on a more physical level we consider each one in turn. First I will describe the parametric oscillator. Consider a figure skater in a spin. As one frequently sees, she is able to increase her rate of spinning by pulling in her arms. Picture the swing from the side so that one sees it swing from left to right and back. Think of the skater's body as being at the top of the swing, you are looking at the top of the skater's head and the skater's body is along the axis passing through the support of the swing. This is a one armed skater whose one arm represents the swing. In her outstretched she holds a heavy ball representing the swinger. The skater pulls her arm in as the "swinger" passes through the lowest point and in the motion. This corresponds to the swinger raising himself at the lowest point of the motion. Just as in the skater's spin, the rotation rate of the swing speeds up. This gives the swing more energy allowing it to swing higher. The lowering of the "swinger" at the ends of the swinging motion, i.e. the extension of the arm, has little effect on the motion since the swing is almost stopped at this point. As this process is repeated the swinging motion is increased. This is the basic mechanism of the parametric oscillator.
For the driven oscillator one must think of the angular motion, or more precisely, the angular momentum. As we picture the swing this is made up of two parts; the motion of the swing as it moves back and forth and the rotation of the swinger as he rotates about the seat of the swing. This is probably clearer for someone seated on the swing. If the swinger rotates suddenly the sum of these two motions is unchanged. But by rotating about the seat in one direction one can impart a rotation of the swing about the support at the top of the swing in the opposite direction. This allows the swinger to give a kick to the motion of the swing about the support and thus drive the swing higher. This is the mechanism of the driven oscillator.
In comparing the two the most readily observable difference is, when the swinger is most active? For the parametric oscillator the swinger raises himself as the swing passes through the lowest point of the arc of the swing's motion. For the driven oscillator the swinger's body is quiet as the swing passes through the lowest point. The swinger simply glides forward or backward through the lowest portion of the arc. As you watch kids or adults at the playground, the second driven oscillator mechanism is clearly what they are doing. Don't just trust me, go see for yourself.
For more information and a detailed discussion see:
W. B. Case and M. A. Swanson, "The pumping of a swing from the seated position", American Journal of Physics 58, 463-467 (1990).
W. B. Case, "The pumping of a swing from the standing position", American Journal of Physics 64, 215-220 (1996).