Pendulum Towing a Rolling Disk
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.[more]
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.[less]
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 .