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Loop the Loop

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This Demonstration simulates the rolling motion of a solid disk inside a ring.

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The disk gains speed while rolling down a ramp of adjustable length and inclination. Inside the ring, the disk either rolls back and forth or makes a full loop inside the ring.

An orange dot-dashed line marks the minimum starting height of the center of the disk to perform a full loop inside the ring.

Only gravity acts on the disk and causes it to roll without slipping.

A gauge shows the potential and kinetic energies during the motion of the disk. The total energy is conserved.

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Contributed by: Erik Mahieu (July 2015)
Open content licensed under CC BY-NC-SA

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Details

The height of the disk center above the bottom line is the piecewise function:

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Here is the angular displacement of the disk, is the length of the ramp, is the inclination of the ramp, is the disk radius, and is the loop radius.

Lagrangian mechanics can be used in this case:

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The resulting equation of motion is:

where

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