## The main task of mechanics

January 3rd, 2012

The main task of mechanics consists of a definition of mathematical kind of dependence of the body position from time:

The solution of the task is possible with a definition of the frame of reference, which consists of a body of reference with associated coordinate system, and of an

agreement about the starting point of time. A movement of the body in the frame of reference is characterized by such concepts as trajectory, path, displacement. Let us consider a body whose dimensions in conditions of movement can be neglected. Such body is called a material point (MP). Let us consider the MP, which has the coordinates (x,y,z).

Fig. 1. The radius vector of the material point and its components

The radius vector of the point M with coordinates (x,y,z) equals to a vector sum of components:

$\vec r = x\vec i + y\vec i + z \vec k$        (2),

and its magnitude is determined according to (3):

$r=\sqrt{x_2+y_2+z_2}$        (3),

The MP magnitude  and the direction of the MP radius vector change during the movement (Figure 2).

Fig.2. The path and the displacement of MP

The vector which connects the starting position of the MP and its position at any given time, directed from the initial position to the position at a given time is called  displacement.

Figure 2 shows that displacement is equal to a difference between two radius vectors of MP: at a given time ( $\vec r$ ) and at the initial time ( $\vec r_0$ ):

$\Delta\vec r =$ $\vec r$$\vec r_0 =$ $(x-x_0)\vec i$ + $(y-y_0)\vec j$ + $(z-z_0)\vec k$           (4)

The locus of points, which MP passes during the movement, is called a trajectory. Depending on the shape of the trajectory there are two types of motion: rectilinear and curvilinear. A path is a number, which is equal to the length of trajectory. It is clear from figure 3 that if the  $\Delta t$ tends to zero then the ratio:

Depending on characteristics of the main task of mechanics solution, it is usually divided to kinematics and dynamics. In the kinematics, we determine mathematical laws of motion without clarifying the reasons leading to the movement. In the dynamics, we consider the laws of motion in view of reasons causing the body movement.

(The information has been taken from e-book «Fundamentals of classical physics” (author V.D. Shvets), which was published in 2007 by “1C-Multimedia Ukraine” publishing company.  Copyright is reserved by certificate Number 55955 from 06.08.2014.More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets/Books  ).