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НАУЧНЫЙ ЖУРНАЛ « IN SITU »
ISSN (p) 2411-7161 / ISSN (e) 2712-9500
№11 / 2023
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Атдаева Айджерен, студентка Пашшагулыева Оразджахан, студентка Ходжамамедов Нурмамет, студент Артыкбаев Бахрам, студент Институт Инженерно-технических и транспортных коммуникаций Туркменистана
TRAJECTORY AND EQUATIONS OF MOTION OF A POINT
Key words:
trajectory, kinematics, mechanics.
The science that studies the laws of motion and equilibrium of solid, liquid, and gaseous bodies is called theoretical (theoretical) mechanics. If anybody in space changes its position with respect to another second body called the frame of reference over time, then that first body is called a body moving with respect to the second body. The concept of linear motion includes the concepts of space, time, moving objects, and computational systems. Broadly, the movement of a body relative to other bodies over time is called mechanical motion. Theoretical mechanics studies the general laws of mechanical motion. Theoretical mechanics is divided into three parts: kinematics, statics, and dynamics
The branch of theoretical mechanics that studies the laws of motion of bodies without regard to the causes that produce and change them is called kinematics. In kinematics it is used as in all theorems and axioms of geometry. Kinematics is a transition from geometry to mechanics because the concept of time is introduced. A body with very small dimensions and finite or infinitesimal mass is called a material point. In the future, instead of calling the material a point, we will call it a point. First, the kinematics of a point, then a system of points, and an absolute solid are studied. All of the mechanics we will use the right coordinate system in the study of sections.
Trajectory and equations of motion of a point. Radius is the codograph of a vector. Point movement detection methods.
Let us consider the motion of point M with radius vector r, taking the axis coordinate system (Fig. 1). The geometric position AMB of the ends of the radius-vectors r1, r2, ..., rn in different positions M1, M2, ..., Mn of this moving point M is called the trajectory of the movement. In this sense, the trajectory of the point's motion is also called the hodograph of its radius vector.
Thus, the line drawn by a point moving in space becomes its trajectory. In general, equations that allow
АКАДЕМИЧЕСКОЕ ИЗДАТЕЛЬСТВО «НАУЧНАЯ АРТЕЛЬ»
us to determine the state and motion (or law of motion) of a point at any point in time are called equations of motion or laws of motion. The motion of the point is determined by three methods: 1. Determination of the motion of the point by the natural method. According to this calculation system, its trajectory is given as the intersection of two cylindrical surfaces and the distance S as a function of time t from the initial point O to the point M on the tractor.
2. Determination of the motion of a point by the method of coordinates. In this method, the coordinates of the moving point are given as a function of time t:
(1.2) These (1.2) are the coordinate equations of motion of a point, and each of them defines the laws of motion of the projections of that point onto the corresponding coordinate axes. Eliminating time t from these equations (1.2), we obtain the following two systems of equations:
(1.3) is the equation of the trajectory of a point given by the intersection of two cylindrical surfaces parallel to the coordinate axes, each of which generates.
3. Determination of the motion of a point by the vector method. In this method, the law of motion of a point is given in vector form. (1.4) When every law is given in vector form, the trajectory of a point is the hodograph of its r-radius-vector. Equation (1.4) is called the vector equation of motion of a point. The point's motion in polar coordinates is r=r(t), eliminating time t from the equations, we get the equation of its trajectory in polar coordinates. (1.5)
Velocity, acceleration of a point and their projections on coordinate axes. Velocity hodograph.
Let's take a right-hand coordinate system and consider the motion of point M with radius vector r. After time t, the radius-vector of the point M is increasing by r. The ratio of the increase in the radius-vector of a moving point to the corresponding increase in time is called the average velocity vector.
From (1. 6 ), we find the actual velocity of the point by passing it to the limit.
(7.1) As we can see, the real velocity vector of a point is equal to the first product of its radius-vector with time and directed tangent to the trajectory at a given point. Denote by n VVV,...,21 the vectors of velocities of the point M moving along the curved line at times t1,t2,...,tn corresponding to the states M1,M2,...,Mn. These velocities are different in magnitude and direction. Let's move them to a point P, called a pole.
As we can see, the projections of the velocity vector of the point on the coordinate axes are equal to the first product of its corresponding coordinates with respect to time. Thus, the modulus of the velocity vector is determined by the following formula.
As can be seen from the expressions, the projections of the acceleration vector of a point on the coordinate axes are equal to the second products of its corresponding coordinates with respect to time. The modulus of the acceleration vector is defined as. The following formulas are used to determine the cosines of the direction angles of the acceleration vector.
Literature:
1. Main National Program of economic, political and cultural development of Turkmenistan for the period up to 2020. Ashgabat, 2003.
2. Bat M.I. Djanelidze G. Yu Kelzon A.S. Theoretical mechanics in examples and problems. volumes I, II, III Moscow 1975.
3. Chetaev. N.G. Theoretical Mechanics Moscow 1978
4. Accounting. N. N Fundamentals of Theoretical Mechanics Part I.II Moscow 1972.
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