# 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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Contributed by: Erik Mahieu (October 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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 .

## Permanent Citation