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The kinetic energy theorem is the work of force. Moscow State University of Printing Arts. Potential energy of interaction of the body with the Earth

The work of the resultant of all forces applied to the body is equal to the change in the kinetic energy of the body.

This theorem is true not only for the translational motion of a rigid body, but also in the case of its arbitrary motion.

Kinetic energy is possessed only by moving bodies, therefore it is called the energy of motion.

§ 8. Conservative (potential) forces.

The field of conservative forces

Def.

Forces, the work of which does not depend on the path along which the body moved, but is determined only by the initial and final positions of the body, are called conservative (potential) forces.

Def.

The field of forces is an area of ​​space, at each point of which a force acts on a body placed there, which regularly changes from point to point in space.

Def.

A field that does not change over time is called stationary.

The following 3 statements can be proved

1) The work of conservative forces on any closed path is 0.

Proof:

2) The uniform field of forces is conservative.

Def.

A field is called homogeneous if at all points of the field the forces acting on the body placed there are the same in magnitude and direction.

Proof:

3) The field of central forces, in which the magnitude of the force depends only on the distance to the center, is conservative.

Def.

The field of central forces is a force field, at each point of which a force acts on a point body moving in it, directed along a line passing through the same fixed point - the center of the field.

In the general case, such a field of central forces is not conservative. If in the field of central forces the magnitude of the force depends only on the distance to the center of the force field (O), i.e. , then such a field is conservative (potential).

Proof:

where is the antiderivative.

§ 9. Potential energy.

The connection between force and potential energy

in the field of conservative forces

Let us choose the origin of coordinates by the field of conservative forces, i.e.

The potential energy of the body in the field of conservative forces. This function is uniquely determined (depends only on coordinates), since the work of the conservative forces does not depend on the type of path.

Let us find the connection in the field of conservative forces when the body moves from point 1 to point 2.

The work of conservative forces is equal to the change in potential energy with the opposite sign.

The potential energy of a body of a field of conservative forces is the energy due to the presence of a force field arising as a result of a certain interaction of a given body with an external body (bodies), which, as they say, creates a force field.

The potential energy of the field of conservative forces characterizes the body's ability to do work and is numerically equal to the work of conservative forces to move the body to the origin (or to a point with zero energy). It depends on the choice of the zero level and can be negative. In any case, and hence for elementary work it is true, i.e. or, where is the projection of the force on the direction of movement or elementary displacement. Hence, . Because we can move the body in any direction, then it is true for any direction. The projection of a conservative force on an arbitrary direction is equal to the derivative of the potential energy in this direction with the opposite sign.

Taking into account the decomposition of vectors and in the basis,, we obtain that

On the other hand, it is known from mathematical analysis that the total differential of a function of several variables is equal to the sum of the products of partial derivatives with respect to the arguments by the differentials of the arguments, i.e. , and hence from the relation we obtain

For a more compact record of these ratios, you can use the concept of the gradient of a function.

Def.

The gradient of some scalar function of coordinates is a vector with coordinates equal to the corresponding partial derivatives of this function.

In our case

Def.

Equipotential surface is the locus of points in the field of conservative forces, the values ​​of potential energy in which are the same, i.e. ...

Because from the definition of an equipotential surface it follows that for points of this surface, then, as the derivative of a constant, hence.

Thus, the conservative force is always perpendicular to the equipotential surface and is directed towards the decrease in potential energy. (P 1> P 2> P 3).

§ 10. Potential energy of interaction.

Conservative mechanical systems

Consider a system of two interacting particles. Let the forces of their interaction be central and the magnitude of the force depends on the distance between the particles (such forces are gravitational and electrical Coulomb forces). It is clear that the forces of interaction between two particles are internal.

Taking into account Newton's third law (), we obtain, i.e. the work of the internal forces of interaction of two particles is determined by the change in the distance between them.

The same work would be done if the first particle was at rest at the origin, and the second received a displacement equal to the increment of its radius vector, i.e. the work done by the internal forces can be calculated by considering one particle motionless, and the second moving in the field of central forces, the magnitude of which is uniquely determined by the distance between the particles. In Section 8, we proved that the field of such forces (i.e., the field of central forces, in which the magnitude of the force depends only on the distance to the center) is conservative, which means that their work can be considered as a decrease in potential energy (determined, according to Section 9, for the field of conservative forces).

In the case under consideration, this energy is due to the interaction of two particles that make up a closed system. It is called the potential energy of interaction (or mutual potential energy). It also depends on the choice of the zero level and can be negative.

Def.

A mechanical system of solids, the internal forces between which are conservative, is called a conservative mechanical system.

It can be shown that the potential interaction energy of a conservative system of N particles is composed of the potential interaction energies of particles taken in pairs, which can be imagined.

Where is the potential energy of interaction of two particles i-th and j-th. The indices i and j in the sum take independent values ​​1,2,3, ..., N. Taking into account that the same potential energy of interaction of the i-th and j-th particles with each other, then when summing up the energy will multiply by 2, as a result of which a coefficient appears in front of the sum. In the general case, the potential interaction energy of a system of N particles will depend on the position or coordinates of all particles. It is easy to see that the potential energy of a particle in the field of conservative forces is a kind of potential energy of interaction of a system of particles, since the force field is the result of some interaction of bodies with each other.

§ 11. The law of conservation of energy in mechanics.

Let a rigid body move translationally under the action of conservative and non-conservative forces, i.e. general case. Then the resultant of all the forces acting on the body. The work is the resultant of all forces in this case.

By the kinetic energy theorem, and also taking into account that, we get

Total mechanical energy of the body

If, then. This is the mathematical record of the law of conservation of energy in mechanics for an individual body.

The formulation of the law of conservation of energy:

The total mechanical energy of the body does not change in the absence of the work of non-conservative forces.

For a mechanical system of N particles, it is easy to show that (*) takes place.

Wherein

The first sum here is the total kinetic energy of the particle system.

The second is the total potential energy of particles in the external field of conservative forces

The third is the potential energy of interaction of the particles of the system with each other.

The second and third sums represent the total potential energy of the system.

The work of non-conservative forces consists of two terms, which represent the work of internal and external non-conservative forces.

As in the case of the motion of an individual body, for a mechanical system of N bodies, if, then, and the law of conservation of energy in the general case for a mechanical system reads:

The total mechanical energy of a system of particles, which are only under the influence of conservative forces, is conserved.

Thus, in the presence of non-conservative forces, the total mechanical energy is not conserved.

Non-conservative forces are, for example, the friction force, the resistance force and other forces, the actions of which cause energy desination (the transition of mechanical energy into heat).

Forces leading to desinication are called desinative. Some forces are not necessarily dissenting.

The law of conservation of energy is universal and is applicable not only to mechanical phenomena, but also to all processes in nature. The total amount of energy in an isolated system of bodies and fields always remains constant. Energy can only pass from one form to another.

Given this equality

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Kinetic energy.

An inalienable property of matter is motion. Various forms of motion of matter are capable of mutual transformations, which, as has been established, occur in strictly defined quantitative ratios. Energy is a single measure of various forms of motion and types of interaction of material objects.

Energy depends on the parameters of the state of the system, ᴛ.ᴇ. such physical quantities that characterize some essential properties of the system. Energy, which depends on two vector parameters that characterize the mechanical state of the system, namely, the radius vector that determines the position of one body relative to the other, and the velocity that determines the speed of movement of the body in space, is called mechanical.

In classical mechanics, it seems possible to split mechanical energy into two terms, each of which depends on only one parameter:

where is the potential energy, depending on the relative location of the interacting bodies; - kinetic energy, depending on the speed of movement of the body in space.

The mechanical energy of macroscopic bodies can only change through work.

Let us find an expression for the kinetic energy of the translational motion of a mechanical system. It is worth saying that first, consider a material point with mass m... Let us assume that its speed at some point in time t is equal. Let's define the work of the resulting force acting on a material point for some time:

Considering that based on the definition of the dot product

where is the initial, and is the final speed of the point.

The quantity

it is customary to call the kinetic energy of a material point.

With the help of this concept, relation (4.12) can be written in the form

From (4.14) it follows that energy has the same dimension as work and, therefore, is measured in the same units.

Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the work of the resultant of all forces acting on a material point is equal to the increment of the kinetic energy of this point. Note that the increment of kinetic energy can be positive or negative, depending on the sign, the work done (the force can either accelerate or slow down the movement of the body). This statement is usually called the kinetic energy theorem.

The result obtained can be easily generalized to the case of translational motion of an arbitrary system of material points. It is customary to call the kinetic energy of a system the sum of the kinetic energies of the material points of which this system consists. As a result of the addition of relations (4.13) for each material point of the system, we again obtain formula (4.13), but already for a system of material points:

where m Is the mass of the entire system.

Note that there is a significant difference between the kinetic energy theorem (the law on the change in kinetic energy) and the law on the change in the momentum of the system. As you know, the increment of the momentum of the system is determined only by external forces. Internal forces, due to the equality of action and reaction, do not change the impulse of the system. This is not the case with kinetic energy. The work of internal forces, generally speaking, does not vanish. For example, when two material points move, interacting with each other by the forces of attraction, each of the forces will do a positive job, and the kinetic energy of the entire system will increase positively. Consequently, the increase in kinetic energy is determined by the work of not only external, but also internal forces.


  • - Kinetic energy theorem

    A curvilinear integral of the 2nd kind, the calculation of which, as a rule, is easier than the calculation of a curvilinear integral of the 1st kind. The power of force is called the work of force per unit of time. Since in an infinitely small time dt the force does the work dA = fsds = fdr, then the power ...

    • The kinetic energy of a material point is expressed by half the product of the mass of this point by the square of its speed.

    The theorem about the kinetic energy of a material point can be expressed in three forms:

    that is, the differential of the kinetic energy of a material point is equal to the elementary work of the force acting on this point;

    i.e. the time derivative of the kinetic energy of a material point is equal to the power of the force acting on this point:

    that is, the change in the kinetic energy of a material point on the final path is equal to the work of the force acting on the point on the same path.

    Table 17. Classification of tasks

    If several forces act on a point, then the right-hand sides of the equations include the work or power of the resultant of these forces, which is equal to the sum of the work or powers of all the constituent forces.

    In the case of rectilinear motion of a point, directing the axis along a straight line along which the point moves, we have:

    where, since in this case the resultant of all forces applied to the point is directed along the x axis.

    Applying the theorem on kinetic energy in the case of non-free motion of a material point, one should keep in mind the following: if a perfect stationary constraint is imposed on the point (the point moves along an absolutely smooth fixed surface or line), then the constraint reaction does not enter the equations, because this reaction is directed along normal to the trajectory of the point and, therefore, its work is zero. If friction has to be taken into account, then the kinetic energy equation will include the work or power of the friction force.

    The tasks related to this section can be divided into two main types.

    I. Problems on the application of the kinetic energy theorem for rectilinear motion of a point.

    II. Problems on the application of the kinetic energy theorem for the curvilinear motion of a point.

    In addition, tasks related to type I can be divided into three groups:

    1) the force acting on the point (or the resultant of several forces) is constant, i.e., where X is the projection of the force (or the resultant) onto the axis directed along the rectilinear trajectory of the point;

    2) the force acting on the point (or the resultant) is a function of the distance (abscissa of this point), i.e.

    3) the force acting on a point (or resultant) is a function of the speed of this point, i.e.

    Type II tasks can be divided into three groups:

    1) the force acting on a point (or resultant) is constant both in magnitude and in direction (for example, the force of weight);

    2) the force acting on a point (or resultant) is a function of the position of this point (a function of the coordinates of the point);

    3) movement of a point in the presence of resistance forces.

    The scalar value T, equal to the sum of the kinetic energies of all points of the system, is called the kinetic energy of the system.

    Kinetic energy is a characteristic of the translational and rotational motion of the system. Its change is influenced by the action of external forces, and since it is a scalar, it does not depend on the direction of movement of the parts of the system.

    Let us find the kinetic energy for different cases of motion:

    1.Translational motion

    The velocities of all points of the system are equal to the velocity of the center of mass. Then

    The kinetic energy of the system in translational motion is equal to half of the product of the mass of the system by the square of the velocity of the center of mass.

    2. Rotational motion(fig. 77)

    The speed of any point of the body:. Then

    or using formula (15.3.1):

    The kinetic energy of a body during rotation is equal to half the product of the moment of inertia of the body relative to the axis of rotation by the square of its angular velocity.

    3. Plane-parallel movement

    With a given movement, the kinetic energy is the sum of the energy of translational and rotational movements

    The general case of motion gives a formula for calculating the kinetic energy, similar to the latter.

    We made the definition of work and power in paragraph 3 of Chapter 14. Here we will consider examples of calculating the work and power of forces acting on a mechanical system.

    1.The work of gravity... Let, the coordinates of the initial and final position of point k of the body. The work of the force of gravity acting on this particle of weight will be ... Then the complete job is:

    where P is the weight of the system of material points, is the vertical displacement of the center of gravity C.

    2. The work of forces applied to a rotating body.

    According to relation (14.3.1), it can be written, but ds according to Fig. 74, due to its infinite smallness, can be represented in the form - infinitesimal angle of rotation of the body. Then

    The quantity called torque.

    Formula (19.1.6) can be rewritten as

    Elementary work is equal to the product of torque and elementary rotation.

    When turning to a final angle, we have:

    If the torque is constant, then

    and the power is determined from the relation (14.3.5)

    as the product of the torque and the angular velocity of the body.

    The theorem on the change in kinetic energy proved for a point (§ 14.4) will be valid for any point of the system

    Composing such equations for all points of the system and adding them term by term, we obtain:

    or, according to (19.1.1):

    which is the expression of the theorem on the kinetic energy of the system in differential form.

    By integrating (19.2.2) we get:

    The theorem on the change in the kinetic energy in the final form: the change in the kinetic energy of the system with some of its final displacement is equal to the sum of the work on this displacement of all external and internal forces applied to the system.

    Let us emphasize that internal forces are not excluded. For an unchangeable system, the sum of the work of all internal forces is zero and

    If the constraints imposed on the system do not change over time, then the forces, both external and internal, can be divided into active and constraint reactions, and equation (19.2.2) can now be written:

    In dynamics, such a concept as an "ideal" mechanical system is introduced. This is such a system, the presence of bonds in which does not affect the change in kinetic energy, that is

    Such connections, which do not change over time and the sum of their work on an elementary displacement is equal to zero, are called ideal, and equation (19.2.5) will be written:

    The potential energy of a material point in a given position M is called a scalar quantity P, equal to the work that the field forces will produce when the point moves from position M to zero

    P = A (mo) (19.3.1)

    Potential energy depends on the position of point M, that is, on its coordinates

    P = P (x, y, z) (19.3.2)

    Let us explain here that a force field is a part of the spatial volume, at each point of which a force determined in magnitude and direction acts on a particle and depends on the position of the particle, that is, on the coordinates x, y, z. For example, the gravitational field of the Earth.

    The function U of coordinates, the differential of which is equal to the work, is called power function... The force field for which a force function exists is called potential force field, and the forces acting in this field are potential forces.

    Let the zero points for two strength functions (x, y, z) and U (x, y, z) coincide.

    By formula (14.3.5) we obtain, i.e. dA = dU (x, y, z) and

    where U is the value of the force function at point M. Hence

    П (x, y, z) = -U (x, y, z) (19.3.5)

    The potential energy at any point of the force field is equal to the value of the force function at this point, taken with the opposite sign.

    That is, when considering the properties of a force field, instead of a force function, one can consider potential energy and, in particular, equation (19.3.3) will be rewritten as

    The work of the potential force is equal to the difference in the values ​​of the potential energy of a moving point in the initial and final positions.

    In particular, the work of gravity:

    Let all the forces acting on the system be potential. Then, for each point k of the system, the work is equal to

    Then for all forces, both external and internal, there will be

    where is the potential energy of the entire system.

    We substitute these sums into the expression for the kinetic energy (19.2.3):

    or finally:

    When moving under the action of potential forces, the sum of the kinetic and potential energy of the system in each of its positions remains constant. This is the law of conservation of mechanical energy.

    A load weighing 1 kg performs free vibrations according to the law x = 0.1sinl0t. Spring stiffness coefficient c = 100 N / m. Determine the total mechanical energy of the load at x = 0.05m, if at x = 0 the potential energy is zero . (0,5)

    A load with a mass of m = 4 kg, dropping down, uses a thread to rotate a cylinder of radius R = 0.4 m. The moment of inertia of the cylinder relative to the axis of rotation I = 0.2. Determine the kinetic energy of the system of bodies at the time when the speed of the load v = 2m / s . (10,5)