Swinging Higher and More Safely: A Mathematical ModelAhn Seonmin

Swinging on swings is one of the classic activities enjoyed by people all over the world. While people swing back and forth automatically, there exist techniques to make one swing higher. We found the optimal method by mathematical modeling. We started off with a model of a simple pendulum and analyzed its motion. We examined 4 realistic methods to swing and concluded that the most efficient method is to pump vigorously after passing the lowest point. We also proposed a safe swing which does not allow one to swing too high. If a swing is propelled excessively and goes too high, it is dangerous. This can be prevented by controlling the damping ratio of a swing. We substituted several damping ratios to our swing model, and found that 0.193 is a safe damping ratio. We used Newton's laws to establish the governing equation for the swing model and considered changes in energy to find the optimal point to start pumping.