Loss of mechanical energy formula. Kinetic and potential energy

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One of the most important laws, according to which a physical quantity - energy is conserved in an isolated system. All known processes in nature, without exception, obey this law. In an isolated system, energy can only change from one form to another, but its amount remains constant.

In order to understand what the law is and where it comes from, let's take a body of mass m, which we drop to the Earth. At point 1, the body is at a height h and is at rest (velocity is 0). At point 2, the body has a certain speed v and is at a distance h-h1. At point 3, the body has a maximum speed and it almost lies on our Earth, that is, h=0

At point 1, the body has only potential energy, since the speed of the body is 0, so the total mechanical energy is equal.

After we released the body, it began to fall. When falling potential energy body decreases, as the height of the body above the Earth decreases, and its kinetic energy increases as the speed of the body increases. In the section 1-2 equal to h1, the potential energy will be equal to

And the kinetic energy will be equal at that moment ( - the speed of the body at point 2):

The closer the body becomes to the Earth, the less its potential energy, but at the same moment the speed of the body increases, and because of this, the kinetic energy. That is, at point 2, the law of conservation of energy works: potential energy decreases, kinetic energy increases.

At point 3 (on the surface of the Earth), the potential energy is zero (since h = 0), and the kinetic energy is maximum (where v3 is the speed of the body at the moment of falling to the Earth). Since , then the kinetic energy at point 3 will be equal to Wk=mgh. Therefore, at point 3 the total energy of the body is W3=mgh and is equal to the potential energy at height h. The final formula of the conservation law mechanical energy will look like:

The formula expresses the law of conservation of energy in a closed system in which only conservative forces act: the total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. There are only mutual transformations of the potential energy of bodies into their kinetic energy and vice versa.

In the formula we used.

Let us summarize the results obtained in the previous sections. Consider a system consisting of N particles with masses . Let the particles interact with each other with forces whose modules depend only on the distance between the particles. In the previous section, we established that such forces are conservative.

This means that the work done by these forces on the particles is determined by the initial and final configurations of the system. Let's assume that apart from internal forces, the i-th particle is affected by an external conservative force and an external non-conservative force . Then the equation i-th movement particles will look like

Multiplying the i-th equation by and adding together all N equations, we get:

The left side represents the increment of the kinetic energy of the system:

(see (19.3)). From formulas (23.14) - (23.19) it follows that the first term on the right side is equal to the decrease in the potential energy of particle interaction:

According to (22.1), the second term in (24.2) is equal to the decrease in the potential energy of the system in an external field of conservative forces:

Finally, the last term in (24.2) is the work of non-conservative external forces:

Taking into account formulas (24.3)-(24.6), we represent relation (24.2) as follows:

Value

(24.8)

is the total mechanical energy of the system.

If there are no external non-conservative forces, the right side of formula (24.7) will be equal to zero and, therefore, the total energy of the system remains constant:

Thus, we have come to the conclusion that the total mechanical energy of a system of bodies, which are acted upon only by conservative forces, remains constant. This statement contains the essence of one of the basic laws of mechanics - the law of conservation of mechanical energy.

For a closed system, i.e., a system whose bodies are not affected by any external forces, relation (24.9) has the form

In this case, the energy conservation law is formulated as follows: the total mechanical energy of a closed system of bodies, between which only conservative forces act, remains constant.

If in a closed system, in addition to conservative forces, there are also non-conservative forces, such as friction forces, the total mechanical energy of the system is not conserved. Considering non-conservative forces as external, in accordance with (24.7) we can write:

Integrating this relation, we get:

The energy conservation law for a system of noninteracting particles was formulated in § 22 (see the text following formula (22.14)).

The law of conservation of mechanical energy: in a system of bodies between which only conservative forces act, the total mechanical energy is conserved, i.e. does not change with time:

Mechanical systems, on the bodies of which only conservative forces (internal and external) act, are called conservative systems.

Law of conservation of mechanical energy can be formulated as follows: in conservative systems, the total mechanical energy is conserved.

The law of conservation of mechanical energy is related to the homogeneity of time. The homogeneity of time is manifested in the fact that physical laws are invariant with respect to the choice of the origin of time reference.

There is another type of system - dissipative systems, in which mechanical energy is gradually reduced due to conversion into other (non-mechanical) forms of energy. This process has been named dissipation (or dissipation) of energy.

In conservative systems, the total mechanical energy remains constant. Only conversions of kinetic energy into potential energy and vice versa can occur in equivalent quantities so that the total energy remains unchanged.

This law is not just a law quantitative conservation of energy, and the law of conservation and transformation of energy, expressing and quality direction of the mutual transformation of various forms of movement into each other.

The law of conservation and transformation of energy - fundamental law of nature, it is valid both for systems of macroscopic bodies and for systems of microscopic bodies.

In a system that also has non-conservative forces, for example, friction forces, total mechanical energy of the system not saved. However, when mechanical energy “disappears”, an equivalent amount of another type of energy always appears.

14. Moment of inertia of a rigid body. moment of impulse. Steiner's theorem.

moment of inertia system (body) relative to a given axis is a physical quantity equal to the sum of the products of the masses of n material points of the system by the squares of their distance to the considered axis:

The summation is performed over all elementary masses m into which the body is divided.

In the case of a continuous distribution of masses, this sum reduces to an integral: where the integration is performed over the entire volume of the body.

The value of r in this case is a function of the position of the point with coordinates x, y, z. Moment of inertia- magnitude additive: the moment of inertia of a body about some axis is equal to the sum of the moments of inertia of the parts of the body about the same axis.

If the moment of inertia of a body about an axis passing through its center of mass is known, then the moment of inertia about any other parallel axis is determined Steiner's theorem:

the moment of inertia of the body J relative to an arbitrary axis is equal to the moment of its inertia Jc relative to a parallel axis passing through the center of mass C of the body, added to the product of the body mass by the square of the distance a between the axes:

Examples of moments of inertia of some bodies (bodies are considered homogeneous, m is the mass of the body):

Angular moment (momentum) material point A relative to a fixed point O is a physical quantity determined by a vector product:

where r is the radius vector drawn from point O to point A;

p = mv - momentum of a material point;

L is a pseudovector, its direction coincides with the direction forward movement of the right screw during its rotation from k.

Momentum vector modulus:

where a is the angle between the vectors r and p;

l is the shoulder of the vector p with respect to the point O.

Angular moment relative to the fixed axis z is called a scalar quantity Lz, equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point O of this axis. The angular momentum Lz does not depend on the position of the point O on the z axis.

When an absolutely rigid body rotates around a fixed axis z, each individual point of the body moves along a circle of constant radius r, with a certain speed Vi. Velocity Vi and momentum mV are perpendicular to this radius, i.e. the radius is the arm of the vector . Therefore, the angular momentum of a single particle is:

Momentum of a rigid body relative to the axis is the sum of the angular momentum of individual particles:

Using the formula, we obtain that the angular momentum of a rigid body about the axis is equal to the product of the moment of inertia of the body about the same axis and the angular velocity:

4.1. Losses of mechanical energy and work of nonpotential forces. K.P.D. Cars

If the law of conservation of mechanical energy were fulfilled in real installations (such as the Oberbeck machine), then many calculations could be done based on the equation:

T O + P O = T(t) + P(t) , (8)

Where: T O + P O = E O- mechanical energy at the initial moment of time;

T(t) + P(t) = E(t)- mechanical energy at some subsequent point in time t.

We apply formula (8) to the Oberbeck machine, where it is possible to change the height of the load on the threads (the center of mass of the rod part of the installation does not change its position). Let's lift the load h from the lower level (where we consider P=0). Let the system with the lifted load first be at rest, i.e. T O = 0, P O = mgh(m is the mass of the load on the thread). After the release of the load in the system, movement begins and its kinetic energy is equal to the sum of the energy of the translational movement of the load and the rotational movement of the rod part of the machine:

T= + , (9)

Where - the speed of forward movement of the load;

, J- angular velocity of rotation and moment of inertia of the rod part

For the moment of time when the load falls to the zero level, from formulas (4), (8) and (9) we obtain:

m gh=
, (10)

Where
,  0k - linear and angular speeds at the end of the descent.

Formula (10) is an equation from which (depending on the conditions of the experiment) it is possible to determine the speed And , mass m, moment of inertia J, or the height h.

However, formula (10) describes ideal type installation, during the movement of parts of which there are no forces of friction and resistance. If the work of such forces is not equal to zero, then the mechanical energy of the system is not conserved. Instead of equation (8), in this case, one should write:

T O +P O = T(t) + P(t) + A s , (11)

Where A s- the total work of non-potential forces for the entire time of movement.

For the Oberbeck machine we get:

m gh =
, (12)

Where ,  k - linear and angular speeds at the end of the descent in the presence of energy losses.

In the installation under study, friction forces act on the axis of the pulley and the additional block, as well as atmospheric resistance forces during the movement of the load and the rotation of the rods. The work of these non-potential forces significantly reduces the speed of movement of machine parts.

As a result of the action of non-potential forces, part of the mechanical energy is converted into other forms of energy: internal energy and radiation energy. At the same time, work As exactly equal to the sum of these other forms of energy, i.e. the fundamental, general physical law of conservation of energy is always fulfilled.

However, in installations where macroscopic bodies move, there are observed mechanical energy loss determined by the amount of work As. This phenomenon exists in all real machines. For this reason, a special concept is introduced: efficiency factor - efficiency. This coefficient determines the ratio of useful work to the stored (consumed) energy.

In the Oberbeck machine, the useful work is equal to the total kinetic energy at the end of the descent of the load on the thread, and the efficiency is is determined by the formula:

efficiency.= (13)

Here P O = mgh- stored energy expended (converted) into the kinetic energy of the machine and into energy losses equal to As, T To- total kinetic energy at the end of the descent of the load (formula (9)).