Total energy and work of a system of particles. Kinetic energy of a system of particles. Kinematics of translational motion
We have shown that the work done to move a particle from position 1 to position 2 can be expressed in terms of the increment in kinetic energy:
In the general case, both potential and nonpotential forces can act on a particle. Thus, the resulting force acting on the particle is:
.
The work of all these forces is used to increase the kinetic energy of the particles:
.
But, on the other hand, the work of potential forces is equal to the decrease in the potential energy of particles:
hence,
The value is called the total mechanical energy of the particle. Let's denote it by E.
Thus, the work of nonpotential forces goes to the increment of the total mechanical energy of the particle.
The increment of the total mechanical energy of a particle in a stationary field of potential forces when moving it from point 1 to point 2 can be written as:
.
If > 0, then the total mechanical energy of the particle increases, and if< 0, то убывает. Следовательно, полная механическая энергия частицы может измениться под действием только непотенциальных сил. Отсюда непосредственно вытекает закон сохранения механической энергии одной частицы. Если непотенциальные силы отсутствуют, то полная механическая энергия частицы в стационарном поле потенциальных сил остается постоянной.
In real processes, where resistance forces act, there is a deviation from the law of conservation of mechanical energy. For example, when a body falls to the Earth, the kinetic energy of the body first increases as the speed increases. The resistance force also increases, which increases with increasing speed. Over time, it will compensate for gravity, and in the future, with a decrease in potential energy relative to the Earth, the kinetic energy does not increase. The work of resistance forces leads to a change in body temperature. The heating of bodies under the action of friction is easy to detect by rubbing the palms together.
The work of the force to move the particle goes to increase the energy of the particle:
dA =( , ) = ( , d ) = (d , )=dE
217. What is bond energy? Explain with the example of the nucleus of an atom.
The binding energy is the difference between the energy of the state in which the constituent parts of the system are infinitely distant from each other and are in a continuous state of active rest and the total energy of the bound state of the system
Where is the total energy of the ith component in the disconnected system, and E is the total energy of the bound system
EXAMPLE:
The nuclei of atoms are strongly bound systems of a large number of nucleons. To completely split the nucleus into its constituent parts and remove them over long distances from each other, it is necessary to expend a certain amount of work A . By bond energy called the energy equal to the work that must be done to split the nucleus into free nucleons
Ebonds = -A
According to the law of conservation, the binding energy is simultaneously equal to the energy that is released during the formation of a nucleus from individual nucleons
What is a macroscopic body, a thermodynamic system?
A macroscopic body is a large body consisting of many molecules.
A thermodynamic system is a set of macroscopic bodies that can interact with each other and other bodies (the external environment) - exchange energy and matter with them.
Why is the dynamic method of description inapplicable to systems consisting of a large number of particles?
It is impossible to apply the dynamic method (to write down the equations of motion and initial conditions for all atoms and molecules and clean out the position of all particles at each moment of time), because to study a system consisting of a large number of atoms and molecules, information must be of a generalized nature and refer not to individual particles, but to the whole set.
What is a thermodynamic method for studying a thermodynamic system?
A method for studying systems of a large number of particles, operating with quantities that characterize the system as a whole (p, V, T) during various energy transformations occurring in the system, without taking into account the internal structure of the bodies under study and the nature of individual particles.
What is a statistical method for studying a thermodynamic system?
A method for studying systems of a large number of particles, operating with regularities and average values of physical quantities characterizing the entire system
What are the basic postulates of thermodynamics?
0: Existence and transitivity of thermal equilibrium:
A and C are in equilibrium with each other, B is a thermometer
The equilibrium state of the thermometer is detected by thermometric parameters.
1: The heat received by the thermodynamic system is equal to the sum of the work of the system on the environment. environment and changes in internal energy.
Q=A+
2: Modern formulation: in a closed system, the change in entropy does not decrease (S ≥ 0)
The increment of the kinetic energy of each particle is equal to the work of all forces acting on the particle: ΔK i = A i . Therefore, work A, which is performed by all forces acting on all particles of the system, when its state changes, can be written as follows: TO, or
(1.6.9)
where K is the total kinetic energy of the system.
So, the increment of the kinetic energy of the system is equal to the work done by all the forces acting on all the particles of the system:
Note that the kinetic energy of a system is an additive quantity: it is equal to the sum of the kinetic energies of the individual parts of the system, regardless of whether they interact with each other or not.
Equation (1.6.10) is valid both in inertial and non-inertial frames of reference. It should only be remembered that in non-inertial reference systems, in addition to the work of interaction forces, it is necessary to take into account the work of inertial forces.
Now let's establish a connection between the kinetic energies of a system of particles in different frames of reference. Let the kinetic energy of the system of particles of interest to us be K in a fixed frame of reference. The speed of the i-th particle in this frame can be represented as, , where is the speed of this particle in a moving frame of reference, a is the speed of the moving system relative to the fixed frame of reference Then the kinetic energy of the system
where is the energy in the moving system, T is the mass of the entire system of particles, is its momentum in the moving reference frame.
If the moving reference frame is connected to the center of mass (C-frame), then the center of mass is at rest, which means that the last term is zero and the previous expression takes the form
where is the total kinetic energy of particles in the C-system, called the self-kinetic energy of the particle system
Thus, the kinetic energy of a system of particles is the sum of its own kinetic energy and the kinetic energy associated with the motion of the system of particles as a whole. This is an important conclusion, and it will be repeatedly used in what follows (in particular, in studying the dynamics of a rigid body).
From formula (1.6.11) it follows that the kinetic energy of the system, particles is minimal in the C-system. This is another feature of the C-system.
The work of conservative forces.
Using formula (1.6.2) and
graphical way of defining work,
Let's calculate the work of some forces.
1.Work done by gravity
The force of gravity is directed
vertically down. Let's choose the z axis,
pointing vertically upwards and
project force onto it.
Let's build a graph
depending on z (Fig.1.6.3). The work of gravity
when moving a particle from a point with a coordinate to a point with a coordinate is equal to the area of the rectangle
As can be seen from the expression obtained, the work of gravity is equal to a change in a certain quantity that does not depend on the particle trajectory and is determined up to an arbitrary constant
2.The work of the elastic force.
The projection of the elastic force on the x-axis indicating the direction of deformation,
