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Sliding on a Parabolic Track

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This Demonstration shows an object sliding with damping on a parabolic track with equation . It explores the effect of the damping coefficient and the parameter on the swing period of the object and the constraining force that keeps it on the track.

Contributed by: Erik Mahieu (November 2013)
With additional contributions by: Franz Brandhuber
Open content licensed under CC BY-NC-SA

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This Demonstration was inspired by a question in the Wolfram Community site, "Simulating Mechanics of a Cylinder Rolling on a Parabola".

Lagrangian mechanics can be used to derive the equations of motion [1]. The potential energy and kinetic energy are repectively

,

,

where is the position of the object at time , is the mass of the object, is the Lagrange multiplier, and is the damping coefficient.

This gives the equations of motion:

,

,

.

Reference

[1] S. Timoshenko and D. H. Young, Advanced Dynamics Chapter III, p. 281, Lagrangian Equations for Impulsive Forces.

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