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Pendulum with Varying Length or How to Improve Your Next Swing Ride

Moving up and down on the seat of the swing influences its motion. This Demonstration simulates a swing as a child crouches down and straightens up from the seat in an oscillating manner, thus mimicking a pendulum with cyclically varying length.
Remember from childhood that to make a swing move higher and higher, you must alternate between crouching and straightening up. The best result is obtained when this is done with a period that is approximately half the period of the swing.
Using the sliders and looking at the curves, you can get ever-increasing swing length, a gracefully periodic motion, or a chaotic movement. Have fun!

SNAPSHOTS

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DETAILS

The system consisting of the swing and the child can be described as a pendulum with varying length. Moving up and down cyclically changes the radial position of the center of gravity and thus the effective length of the pendulum.
is the length of the swing, and are the amplitude and frequency of the radial movement of the gravity point, and are the radial and angular positions of the center of gravity, is the mass of the child, and is the force of gravity.
Lagrangian mechanics can be used to model the system.
The effective length of the swing is .
The potential energy is .
The kinetic energy is .
The Lagrangian is
and the resulting equation of motion is .
snapshot 1: the classical way to get a swing to go higher and higher is to stand up at highest points of each swing (forward and backward) and to crouch at the bottom, with a frequency approximately double the frequency of the swing (see bookmark "expert")

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