Kinematics of material points

Pagina 1

Note: In the following, vectors are written in boldface. Contrary to the common notation used in ma thematics, a generic vector AB , with origin in A and endpoint (tip) in B, will be denoted ( B – A ) instead of ( A – B ). In fact, if x A, yA, zA are the coordinates of the point A and xB, yB, zB are the coordinates of the point B, the vector AB can be expressed as: ( ) ( ) ( ) ( )BA B A BABA xx yy zz= −= −+ −+ −AB i jk 1. KINEMATICS OF A MATERIAL POINT 1.1 Absolute reference systems Let us consider an absolute reference system represented by a set of Cartesian axes, x -y (two-dimensional system, 2D) or x -y-z (three-dimensional system, 3D). Let us denote O the origin of the axes. Being the reference system absolute the axes are fixed, that is the origin O does not move and the axes do not rotate. For the sake of simplicity, first let us consider a two -dimensional case study. The positive versus of the axes x and y is indicated by the versors (unit or basis vectors) i and j. Since the axes of the reference system do not rotate, the versors i and j are constant. Therefore, their derivatives with respect to time are null: 0t d d =i 0t d d =j (1.1.1) In the case of a point that moves in a 3D space the positive versus of the axes x -y-z is indicated by the versors i , j and k. 1.1.1 Absolute position, velocity and acceleration of a material point Let us consider a material point that moves in the x-y plane determined by an absolute reference system of Cartesian axes. The loci curve of the time -varying position of the point P(x,y) is the trajectory, s, which can be a general curve or a straight line (Figure 1.1.1). In general, the trajectory can be expressed by means of a function y = f (x), or by z = f (x, y), in the case of a point that moves in a 3D space. Being the…