At the bottom of the swing, the pumper raises herself by pulling quickly backward on the ropes. At that point there is no applied external torque, so conservation of angular momentum applies, and mv(1)r(1)=mv(2)r(2), where (1) refers to before the pull and (2) refers to after the pull. Pulling backward on the ropes raises the pumper, decreasing the radius with respect to thesupport point, and thus increasing her velocity. By conservation of energy the pumper thus rises to a higher level with each pump.
This analysis is perhaps somewhat of an oversimplification, but may be adequate for the beginner. See some of the references for a more complete and mathematically detailed description. This can only be done with a small subject using the fork truck swing.