TRAJECTORIES OF POINTS OF A FLAT FIGURE, A CURVILINE CIRCUIT WHICH ROLLS WITHOUT SLIDING ON A DIRECT LINE

Abstract

The plane-parallel motion of a figure bounded by a curvilinear contour is considered. The figure rolls without sliding on a straight line. An analytical description of finding the trajectory of a point, which is rigidly fixed to the figure, or makes a relative motion in the moving system of the figure, is developed. A classic example of this is finding a trajectory of a point on the rim of a wheel that rolls without sliding in a straight line. It is well known that such a trajectory is a cycloid. If the point is located outside the rim, then its trajectory is an elongated cycloid, if inside the rim is a shortened cycloid. This list exhausts the set of possible trajectories of a shape in the form of a wheel. For second-order curves that act as a curvilinear contour of a figure that rolls on a straight line, the set of trajectories of the individual points of the figure increases. If such a point is the focus of the second-order curve, then its trajectory is a known curve which name is assigned to the specific name of the curve. In particular, if such a curve is a parabola, then the trajectory of focus is a known curve, called a chain line. The choice of a point in an arbitrary place of a curvilinear contour, bounded by a parabola, expands the set of trajectories. The paper presents generalized parametric equations for finding them if the curve of a curvilinear contour is given by parametric equations. Specific examples for a flat figure bounded by a parabola are considered. It is shown that the trajectory of the focus of the parabola, which rolls on a straight line, is a chain line. The circle was checked and the known curve - cycloid, and its variants - elongated and shortened cycloid were obtained. A case is considered when a point of a flat figure makes relative movement with respect to the internal coordinates of a moving flat figure. To find the absolute trajectory of a point, the figurative motion of a flat figure, which is rolled in a straight line, and the relative motion in a moving coordinate system rigidly connected to the flat figure, taken into account. The paper focuses on curves of curvilinear contour, in parametric equations of which the arc length of the curve acts as an independent variable. This is important because this parameter is crucial in the analytical description of the rolling of a flat figure in a straight line. The analytical description is based on the equality of the length of the arc of the contour of the flat figure and the line along which it is rolling. For simplify of the analytical calculations the curves of contour given by the natural equation are considered. Examples are given and the trajectories of points of a flat figure are constructed, outlined with a contour which curve is given by the known natural equation of the chain line.