Torsional Pendulum Suspended by Two Stranded Wires

This Demonstration simulates a variation of the classic torsional pendulum.
A disk hangs on a strand of two wires. The disk is given an initial angular displacement and released from rest, resulting in a harmonic motion similar to that of a torsional pendulum.
The potential energy of this system changes cyclically due to the variation in length of the strand, from the twisting and untwisting of the wires.
The changes in potential energy are compensated by changes in the kinetic energy of the rotating disk, with the total energy of the system remaining constant.
Friction between the wires and air drag are not considered.

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DETAILS

This system has one degree of freedom, , the angular displacement of the suspended disk at time .
The potential energy of the system is .
is the effective length of the strand after twisting by the angle : , where
is the length of the untwisted strand, and
is the radius of the strand or the distance between the wires.
The kinetic energy of the system is , where
is the mass of the disk,
is the radius of the disk,
is the width of the disk, and
is its density.
The Lagrangian of this system is .
Substituting this in the Euler–Lagrange equations for gives:
,
.
This results in the equation of motion:
.

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