Dynamics. Dynamics of mechanical particle and particle systems (many body systems)

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2 Dynamics Dynamics of mechanical particle and particle systems (many body systems)

3 Newton`s first law: If no net force acts on a body, it will move on a straight line at constant velocity or will stay at rest if it stays initially at rest. Inertial frame of reference: The frame of reference in which Newton`s first law is valid is called inertial frame of reference. Momentum: Product of the mass and the velocity vector of the mechanical particle.

4 Newton`s second law: If a net external force acts on a body, then the body accelerates. The mass of the body multiplied by the acceleration vector of the body equals the net force vector. Momentum theorem: The first time derivate of the momentum vector of the body equals to the net force acts on the body.

5 Newton`s third law ( Action-reaction law ): If a body exerts a force on a second body, then the second body exerts a force on the first one. The two forces have the same magnitude, but they show in opposit direction. (The have same same direction but different derectionality). Action and reaction are always equal and opposite.

6 Newton`s fourth law ( Superposition law ): If there are several forces acting simultaneously on a body, the net force equals to the vector sum of the acting forces. Each of the forces exerts on the body as it would be alone.

7 Fundamental equation of dynamics: If there are several forces acting simultaneously on a body, the net force equals to the product of the mass and the acceleration vector of the body. In this case the net force holds all information from the neighbor (environment) of the body. Belongs to the body Belongs to the environment Conservation of momentum theorem: If the net force acts of the body is zero, then the momentum of the body is constant. It directly comes out from momentum theorem, if the force is zero.

8 Work: a.) Consider a body that undergoes a displacement of magnitude s along a straight line. While the body moves a constant force acts on it in the same direction as the displacement shows. b.) If the force and the displacement do not parallel to each other, then the work can be defined as: follows: c.) In general case:

9 Energy: Kinetic energy: E = 1 2 mv2 Work-energy theorem: The work done by the net force on a body equals the change in the kinetic energy of the body. Work of the gravitational force in the field of gravity:

10 Work of the gravitational force is independent of the path taken from A to B. It depends only on the coordinates of the two end points (initial and final points) of the path. Conservative force: If the force force is independent of the path taken from A to B. It depends only on the coordinates of the two end points (initial and final points) of the path. The gravitational force is conservative force. The field of gravity is a conservative field. Potential energy:

11 Conservation of mechanical energy theorem: In a conservative field the total mechanical energy (the sum of the potential and kinetic energy) is constant. It means the total energy is conserved. Consequence: If the body moves around a closed path (closed loop), the total work done by the conservative force is always zero. Nonconservative force = dissipative force, an example: Kinetic frictional force:, where is perpendicular to the surface

12 Power instantaneous power: The first time derivate of the work is defined as instantaneous power: SI unit of the dimension: In a special case:

13 Dinamics of the circular motion Single Mechanical Particle Torque and angular momentum Torque or moment: Torque or moment is a physical quantity gives angular acceleration for a body. It has an important role at the rotational motion of a rigid body (see later.) Torque is the moment of a force relative to point O: : lever arm In the English literature the name moment is usually used as term of the moment of force. Torques is usually used for the net moment due to external forces in systems where the vector sum of these forces is zero.

14 Angular momentum The analog quantity of linear momentum (momentum) of a mechanical particle is called as angular momentum of a particle relative to a given point. Angular momentum theorem: The rate of change of angular momentum of a particle equals to the moment or torque of the net force acts on it. In other words: The first time derivate of the angular momentum gives the torque or moment of the net force acts on it. Principle of conservation of angular momentum theorem: If the torque or moment of the net force acts on the body is zero, then the angular momentum of the body is constant.