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## Glossary

# Pumping swing

You remember that a way to increase the amplitude of the swing oscillations is to perform a characteristic “up-and-down” movement of the center-of-mass of your body as the swing oscillates. You do that twice every period: going forward, and also backward. This corresponds to a driving period T = T

_{0}/2, where T

_{0}is the natural period T

_{0}= 2π/ω

_{0}, hence a frequency Ω

_{1}= 2ω

_{0}. But when you were a little boy, your father would push you, usually once every period, hence with Ω

_{2}= ω

_{0}. More generally, you could drive the system by “pushing” only once everyn half-periods, hence with a driving frequency

\[
\Omega_n = \frac{2\omega_0}{n} = \frac{2}{n} \sqrt{\frac{g}{\ell}} , \qquad \mbox{with} \quad n= 1,2,3,\ldots .
\]

These are the “resonant” driving frequencies of the ordinary pendulum, where ω_{0}is its natural frequency of pendulum with length ℓ.

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