Constructing a meridian for a given motion of a particle on a surface rotating around a vertical axis

Abstract

The article deals with the design of a surface of rotation, which is reduced to finding its meridian according to specified conditions. These conditions are the nature of the particle's motion on the inner surface when it rotates around a vertical axis. The absolute motion of a particle is formed from the ratio of the rotational motion of the surface and the relative motion (sliding) of the particle on the surface. Classical examples of such motion are the motion of a particle inside a vertical cone rotating with a constant angular velocity around its axis, as well as the special case when the angle of inclination of the cone's constituent parts is zero and it turns into a horizontal disk. The meridian curve can be given by an explicit equation or by parametric equations as a function of the independent variable. In this article, we consider the case when the meridian of a surface of rotation is given by parametric equations as a function of time. This makes it possible to compose a differential equation of motion of a particle in which all dependencies are functions of time. These dependencies need to be found from the compiled differential equation of motion of the particle. When a particle hits a surface, it starts to slide along it, describing a curved trajectory. Taking into account the rotational motion of the surface, the absolute trajectory is found. The first derivative of its length in time gives the absolute velocity, and the second gives the absolute acceleration, the expression of which includes the unknown functions describing the meridian. The differential equation of motion is written in projections on the three axes of the Cartesian coordinate system. The system of three equations includes four unknown functions: two equations describing the meridian, the dependence of the angular velocity of the particle sliding, and the surface reaction. To solve the equation, the number of unknown functions must be reduced to three. To do this, we define one dependence. This approach leads to special cases, one of which is the movement of a particle on a horizontal disk rotating around a vertical axis. A specific example is considered and a meridian curve is constructed as a result of numerical solution of the equations, provided that the particle inside the surface rises upward with a constant given speed.