On the Equilibria of a Heavy Hoop Suspended on a Nail

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We consider the plane problem of the equilibrium of a homogeneous heavy thin elliptical hoop suspended on a thin horizontal nail. Under the assumption that a dry friction force acts between the nail and the hoop, the dependence of the set of equilibrium positions on the friction coefficient and the semi-axes of the ellipse is studied. The results obtained apply to the following problem: to describe the equilibrium positions of a heavy solid (“gun”) suspended on a nail using a rope, both ends of which are fixed in the body. It is shown how such a distribution of results can be carried out directly in the case when the center of mass of the body is located in the middle between the suspension points.

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Sobre autores

A. Burov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Autor responsável pela correspondência
Email: jtm@narod.ru
Rússia, Moscow, 119333

V. Nikonov

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences

Email: nikon_v@list.ru
Rússia, Moscow, 119333

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2. Fig. 1. A heavy homogeneous rough hoop in the form of an ellipse with semi-axes a and b, suspended on a nail P

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3. Fig. 2. Partial cases of equilibria at μ = 0

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4. Fig. 3. Families of non-isolated equilibria (grey)

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5. Fig. 4. The dark region on the parameter plane (p, μ) corresponds to the points for the values of which the hoop will be in equilibrium when suspended by any of its points

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6. Fig. 5. The dark area on the parameter plane (q, μ) corresponds to the points for the values of which the shotgun will be in equilibrium when suspended by any strap

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7. Fig. 6. Illustration of a mechanical system: a gun suspended on a belt and positioned in a vertical plane (left). Diagram of forces and coupling reactions (right)

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