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Newtonian Hoop applet

This applet models a Newtonian rotating elastic hoop as a polygon with point masses at the vertices, and edges consisting of elastic material; the edges are assumed to have negligible mass, always to be straight line segments, and to obey Hooke’s law exactly. (Note that only one in every 10 vertices is marked, rather than all of them; this gives a clearer sense of the motion of the hoop by avoiding “strobe” effects.)

The applet allows a choice for:

• the number of vertices;
• the constant K, equal to the speed of sound in the relaxed material divided by √2;
• the angular velocity, ω, of the hoop in its equilibrium state;
• the shape of an initial perturbation away from the equilibrium state.

For a given value of K, ω must be less than K√2 in order for an equilibrium state to exist. When a choice is made for K, ω is initially set close to the maximum possible value, but it can then be made lower.

The hoop is drawn colour-coded by tension. Sketched in grey are a circle showing the relaxed radius of the hoop; another circle showing the equilibrium radius of the hoop; and short radial marks showing all locations where the hoop crosses the equilibrium radius.

The choices for the initial perturbation include an axially symmetric change in the radius, which will trigger pulsations of the hoop, and examples of all the modes discussed in the section on Newtonian vibrations in the accompanying theoretical treatment of pulsations and vibrations. The parameter c referred to in the choices is the fraction of ω at which vibrations travel (in the limit of infinitesimal perturbations), relative to the material of the hoop. As expected, all the c=–1 cases are essentially stationary in the non-rotating frame in which the hoop is displayed, because the wave is travelling backwards through the material at exactly the same speed at which the hoop is rotating. For the other choices, c will be one of three solutions to a cubic equation, the coefficients of which depend on the values of K and ω.

All of the perturbations offered are stable in the infinitesimal limit, but since the applet models finite perturbations, if ω is lowered so that the hoop is close to its relaxed state then these modes can result in the hoop “crumpling”.

As with all numerical simulations, the results are subject to cumulative errors. As a guide, the applet constantly calculates the total energy and angular momentum of the hoop, and reports the percentage (the worst of the two cases) by which either conserved quantity has drifted away from its initial value due to numerical errors.  Science Notes / Newtonian Hoop applet / created Monday, 18 June 2007 / revised Wednesday, 20 June 2007