Dynamics of a hopping hoop with a massive point on an absolutely rough plane.

In this article, the well-known problem of the «hopping hoop», which was incorrectly solved by J. Littlewood in his book «A Mathematician’s miscellany» [1], is solved in a more general formulation. The conditions connecting the hoop mass, its radius, the mass of the point and its initial velocity under which the plane reaction force is positive, and, therefore, a jump is impossible, are found in analytical form. A condition is found under which hopping occurs at the initial moment of time, when the point is at the top of the hoop. It is shown that the rolling model with an absolutely rigid and rough hoop and plane at a certain ratio of parameters is incorrect and leads to a contradiction called a «quasijump». It was also noted that the «skimming» phase of the motion, described in the work of W. F. D. Theron and N. M du Plessis «The dynamics of a massless hoop» [6], is not derived, but wrongly postulated.