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Pendulum Towing a Rolling Disk

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This Demonstration simulates the motion of a double pendulum with a rolling disk attached to the end of its lower rod. The disk is free to roll without friction over a horizontal plane.

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Lagrangian mechanics are used to determine the equations of motion. You can change the geometrical parameters using the sliders.

A phase curve gives an overview of the state of the system. A horizontal energy gauge shows the total energy of the system, which remains constant while the potential and kinetic energies change in opposite directions.

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Contributed by: Erik Mahieu (October 2014)
Open content licensed under CC BY-NC-SA

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Details

The pendulum's upper and lower rods have lengths and and angular positions and . The upper bob has mass . The disk has angular position , mass , and radius .

This system has only one degree of freedom: . The other variables are fixed by the algebraic constraints:

,

.

The potential energy of the disk and pendulum system is .

The kinetic energy is .

The Lagrangian of this system is . Substituting this in the Euler–Lagrange equations for and gives

,

,

which results in the equation of motion:

,

where and .

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