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:

Consequently,

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 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 the system is an additive quantity: it is equal to the sum of the kinetic energies separate parts systems, 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 also 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 equal to K in a fixed frame of reference. The speed of the i-th particle in this frame can be represented as 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,

The value that equates to half of the product of the mass of a given body and the speed of this body squared is called in physics the kinetic energy of the body or the energy of action. The change or inconstancy of the kinetic or driving energy of the body for some time will be equal to the work that has been done for a given time by a certain force acting on a given body. If the work of any force along a closed trajectory of any type is equal to zero, then a force of this kind is called a potential force. The work of such potential forces will not depend on the trajectory along which the body moves. Such work is determined by the initial position of the body and its final position. The starting point or zero for the potential energy can be chosen absolutely arbitrarily. The value that will be equal to the work done by the potential force to move the body from a given position to the zero point is called in physics the potential energy of the body or the energy of the state.

For various kinds forces in physics, there are various formulas for calculating the potential or stationary energy of a body.

The work done by potential forces will be equal to the change in this potential energy, which must be taken in the opposite sign.

If you add the kinetic and potential energy of the body, you get a value called the total mechanical energy of the body. In a position where a system of several bodies is conservative, the law of conservation or constancy of mechanical energy is valid for it. A conservative system of bodies is such a system of bodies that is subject to the action of only those potential forces that do not depend on time.

The law of conservation or constancy of mechanical energy is as follows: "During any processes that occur in a certain system of bodies, its total mechanical energy always remains unchanged." Thus, the total or all mechanical energy of any body or any system of bodies remains constant if this system of bodies is conservative.

The law of conservation or constancy of total or all mechanical energy is always invariant, that is, its form of writing does not change, even when the starting point of time is changed. This is a consequence of the law of homogeneity of time.

When dissipative forces begin to act on the system, for example, such as, then a gradual decrease or decrease in the mechanical energy of this closed system occurs. This process is called energy dissipation. A dissipative system is a system in which the energy can decrease over time. During dissipation, the mechanical energy of the system is completely converted into another. This is fully consistent with the universal law of energy. Thus, there are no completely conservative systems in nature. One or another dissipative force will necessarily take place in any system of bodies.

It is known that the increment of the kinetic energy of a particle when moving in a force field is equal to the elementary work of all forces acting on the particle: . If a particle is in a stationary field of conservative forces, then, in addition to the conservative force, other forces, called external ones, can act on it; Then the resulting force is: .

The work of all these forces goes to change the kinetic energy of the particle:

It is also known that the work of conservative field forces can be written as a decrease in the potential energy of a particle in this field.

So or

That. the work of external forces goes to the increment of the value . This value is called full mechanical energy particles in the field: .

From this it can be seen that is determined up to a constant, since , is determined up to a constant. Now you can write

i.e., the increment of the total mechanical energy of a particle on a certain path is equal to the work of external forces acting on the particle along this path; If , then the total mechanical energy of the particle increases. When - decreases.

Example: For a body falling from a cliff, the work of external forces:

Where are the resistance forces.

End of work -

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Kinematics of translational motion

Physical foundations of mechanics.. kinematics of translational motion.. mechanical motion as a form of existence..

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