Download Dynamics of particle systems (many body system)

Dynamics
Single Mechanical Particle
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.
Particle Systems (Many Body Systems)
Single Mechanical Particle
Particle Systems (Many Body Systems)
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.
Momentum of the system – Momentum theorem:
The first time derivate of the net momentum vector of
the many body system equals to the vector sum of all
external forces act on the particles of the system.
Single Mechanical Particle
Particle Systems (Many Body Systems)
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).
Internal forces, external forces:
There are forces acting between particles. Forces can be:
- External forces (acting from external outside source(s))
- Internal forces (acting between particles of the
system)
Action and reaction are always equal and opposite.
Single Mechanical Particle
Particle Systems (Many Body Systems)
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.
Center of mass:
Center of mass is defined as the point with displacement
vector of 𝑟𝑐 relative to O, where the net moment of the
gravitational force exerted on the particles of the system
is zero.
Each of the forces exerts on the body as it would be
alone.
Single Mechanical Particle
Particle Systems (Many Body Systems)
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.
External forces and motion of center of mass:
Center of mass theorem:
Belongs to the
body
Belongs to the
environment
If
then:
Center of mass moves as a point with mass of M and
acceleration of 𝑎𝑐 .
Conservation of center of mass:
If the net external force is zero, then the velocity of center
of mass is constant. In other words: Velocity of center of
mass of an isolated system is constant.
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.
Conservation of momentum theorem:
If no net external force acts on a particle system, then the
total momentum of the whole system remains constant.
This is valid in isolated system.
Single Mechanical Particle
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:
Particle Systems (Many Body Systems)
Single Mechanical Particle
Particle Systems (Many Body Systems)
Energy:
Collisions:
Kinetic energy:
1
𝐸 = 2 𝑚𝑣 2
Work-energy theorem:
The work done by the net force on a body equals the
change in the kinetic energy of the body.
In any collision in which the external force can be
neglected, the momentum is conserved.
a.) Completely inelastic collision:
Total kinetic energy after the collision is less than
before. The conservation of momentum gives the
relationship:
Work of the gravitational force in the field of
gravity:
b.) Elastic collision:
The kinetic energy and the momentum is conserved.
Single Mechanical Particle
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:
Particle Systems (Many Body Systems)
Single Mechanical Particle
Particle Systems (Many Body Systems)
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
perpendicular to the surface
is
Single Mechanical Particle
Power – instantaneous power:
The first time derivate of the work is defined as
instantaneous power:
SI unit of the dimension:
In a special case:
Particle Systems (Many Body Systems)
Dinamics of the circular motion
Single Mechanical Particle
Particle Systems (Many Body Systems)
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.
Single Mechanical Particle
Particle Systems (Many Body Systems)
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 of a particle systems:
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.
Angular momentum theorem:
The net moment of the external forces acting on a
particle system causes the change of the total
angular momentum of the system. In other words:
The first time derivate of the sum of the angular
momentum vector of the system gives the total net
moment vector of the system caused by all external
forces.
Principle of conservation of angular momentum
theorem:
If the net moment due to external forces acting on a
particle system is zero, then the total angular
momentum of the system remains constant.