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Based on my education in biomechanics and from looking at research on standing vertical jumping and on the jumps (high, long, triple) in track and field, I have concluded that the arm circle accomplishes three things for the gymnast.
Those three things are the following:
1) The backwards arm circle slows the forward rotation of the upper body so that the center of gravity does not move too far in front of the feet upon touchdown with the floor and cause too much rotation and a lack of height. Furthermore, this should assist in positioning the center of gravity slightly behind the feet and allow for more time in contact with the floor as the center of gravity pivots over top of the feet just prior to take-off. This is useful in that it allows for more time for force to be applied to the floor and creates a greater impulse. (impulse = force x time)
2) As the arms swing up, they eventually stop and the angular momentum contained within them is then transferred to to the rest of the body in the same direction.
3) The upward swing of the arms actually creates a downward reaction force against the ground. If you stand up and swing your arms backwards/upwards really fast and stop them around shoulder/eye level, you should feel a downward push against the ground.
So, those are the advantages of the arm circle as I've been able to uncover. I feel that they apply both to vault and floor.
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