Pendulum with Counterweight

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A rod is supported in the middle (which is taken as the origin) and can turn freely. On one end of the rod there is a pivot with a pendulum attached to a red bob swinging from it. At the other end of the rod is a blue counterweight.

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Lagrangian mechanics is used to compute the equations of motion of the counterweight and the pendulum and to plot the trace of the pendulum bob.

A horizontal energy gauge shows the total energy of the system, which remains constant as the potential and kinetic energy magnitudes change in opposite directions.

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


Snapshots


Details

The system as described has two degrees of freedom: and are the angular displacements of the counterweight arm and the pendulum, with initial values and .

The rod length is and the counterweight has mass The pendulum has rod length and its bob has mass .

The potential energy of the system is .

The kinetic energy of the system is .

The resulting Lagrangian is and the equations of motion are

,

.



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