In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.

An algorithm for the investigation of a mathematical model of the impact of a double pendulum against an obstacle is constructed and realized by computer. This algorithm allows calculation of impact loads and coefficients of restitution at the contact point and hinges. The main goal of this investigation is to minimize negative impact conditions in the hingings.

The problem ofšdynamics ofšheavy uniform ball moving onšthe fixed rough plane under its own inertia and forces ofšdry friction isšconsidered. Assuming that friction coefficient isšvariable, the switching curve for change the value ofšfriction coefficient isšconstructed. Using this curve tošchange the value ofšfriction coefficient wešhave shown that the bundle ofšequal balls starting from one interval with equal linear and angular velocities should gather atšone point.

The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.