Total mechanical energy of a particle. Law of conservation of total mechanical energy of a particle Kinematics of translational motion

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 a body over time will be equal to the work that has been done in given time 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. Reference point or zero for potential energy can be chosen completely 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 the system of several bodies is conservative, the law of conservation or constancy is valid for it mechanical energy. 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.

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 with other bodies ( external environment) to 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)

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 -

This topic belongs to:

Kinematics of translational motion

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

If you need additional material on this topic, or you did not find what you were looking for, we recommend using the search in our database of works:

What will we do with the received material:

If this material turned out to be useful for you, you can save it to your page on social networks:

tweet

All topics in this section:

mechanical movement
Matter, as is known, exists in two forms: in the form of substance and field. The first type includes atoms and molecules, from which all bodies are built. The second type includes all types of fields: gravity

Space and time
All bodies exist and move in space and time. These concepts are fundamental to all natural sciences. Any body has dimensions, i.e. its spatial extent

Reference system
To unambiguously determine the position of a body at an arbitrary point in time, it is necessary to choose a reference system - a coordinate system equipped with a clock and rigidly connected to an absolutely rigid body, according to

Kinematic equations of motion
When t.M moves, its coordinates and change with time, therefore, to set the law of motion, it is necessary to specify the type of

Movement, elementary movement
Let point M move from A to B along a curved path AB. At the initial moment, its radius vector is equal to

Acceleration. Normal and tangential accelerations
The movement of a point is also characterized by acceleration - the speed of change in speed. If the speed of a point in an arbitrary time

translational movement
The simplest form of mechanical motion of a rigid body is translational motion, in which the straight line connecting any two points of the body moves with the body, remaining parallel | its

Law of inertia
Classical mechanics is based on Newton's three laws, formulated by him in the work "Mathematical Principles of Natural Philosophy", published in 1687. These laws were the result of a genius

Inertial frame of reference
It is known that mechanical motion is relative and its nature depends on the choice of reference frame. Newton's first law is not valid in all frames of reference. For example, bodies lying on a smooth surface

Weight. Newton's second law
The main task of dynamics is to determine the characteristics of the motion of bodies under the action of forces applied to them. It is known from experience that under the influence of force

The basic law of the dynamics of a material point
The equation describes the change in the motion of a body of finite dimensions under the action of a force in the absence of deformation and if it

Newton's third law
Observations and experiments show that the mechanical action of one body on another is always an interaction. If body 2 acts on body 1, then body 1 necessarily counteracts those

Galilean transformations
They allow one to determine the kinematic quantities in the transition from one inertial frame of reference to another. Let's take

Galileo's principle of relativity
The acceleration of any point in all frames of reference moving relative to each other in a straight line and uniformly is the same:

Conserved quantities
Any body or system of bodies is a collection of material points or particles. The state of such a system at some point in time in mechanics is determined by setting the coordinates and velocities in

Center of mass
In any system of particles, you can find a point called the center of mass

Equation of motion of the center of mass
The basic law of dynamics can be written in a different form, knowing the concept of the center of mass of the system:

Conservative forces
If a force acts on a particle placed there at each point in space, it is said that the particle is in a field of forces, for example, in the field of gravity, gravitational, Coulomb and other forces. Field

Central Forces
Any force field is caused by the action of a certain body or system of bodies. The force acting on a particle in this field is about

Potential energy of a particle in a force field
The fact that the work of a conservative force (for a stationary field) depends only on the initial and final positions of the particle in the field allows us to introduce the important physical concept of potentially

Relationship between potential energy and force for a conservative field
The interaction of a particle with surrounding bodies can be described in two ways: using the concept of force or using the concept of potential energy. The first method is more general, because it applies to forces

Kinetic energy of a particle in a force field
Let a particle with mass move in forces

Law of conservation of mechanical energy of a particle
It follows from the expression that in a stationary field of conservative forces, the total mechanical energy of a particle can change

Kinematics
Rotate the body through some angle

The angular momentum of the particle. Moment of power
In addition to energy and momentum, there is another physical quantity with which the conservation law is associated - this is the angular momentum. Particle angular momentum

Moment of momentum and moment of force about the axis
Let us take in the frame of reference we are interested in an arbitrary fixed axis

The law of conservation of momentum of the system
Let us consider a system consisting of two interacting particles, which are also acted upon by external forces and

Thus, the angular momentum of a closed system of particles remains constant, does not change with time
This is true for any point in the inertial frame of reference: . Angular moments of individual parts of the system m

Moment of inertia of a rigid body
Consider a rigid body that can

Rigid Body Rotation Dynamics Equation
The equation of the dynamics of rotation of a rigid body can be obtained by writing the equation of moments for a rigid body rotating around an arbitrary axis

Kinetic energy of a rotating body
Consider an absolutely rigid body rotating around a fixed axis passing through it. Let's break it down into particles with small volumes and masses

Work of rotation of a rigid body
If a body is rotated by a force

Centrifugal force of inertia
Consider a disk that rotates with a ball on a spring, put on a spoke, Fig.5.3. The ball is

Coriolis force
When a body moves relative to a rotating CO, in addition, another force appears - the Coriolis force or the Coriolis force

Small fluctuations
Consider a mechanical system whose position can be determined using a single quantity, say x. In this case, the system is said to have one degree of freedom. The value of x can be

Harmonic vibrations
The equation of Newton's 2nd Law in the absence of friction forces for a quasi-elastic force of the form has the form:

Mathematical pendulum
This is a material point suspended on an inextensible thread with a length that oscillates in a vertical plane.

physical pendulum
This is a rigid body that oscillates around a fixed axis associated with the body. The axis is perpendicular to the drawing and

damped vibrations
In a real oscillatory system, there are resistance forces, the action of which leads to a decrease in the potential energy of the system, and the oscillations will be damped. In the simplest case

Self-oscillations
With damped oscillations, the energy of the system gradually decreases and the oscillations stop. In order to make them undamped, it is necessary to replenish the energy of the system from the outside at a certain moment

Forced vibrations
If the oscillatory system, in addition to the resistance forces, is subjected to the action of an external periodic force that changes according to the harmonic law

Resonance
The curve of the dependence of the amplitude of forced oscillations on leads to the fact that for some specific for a given system

Wave propagation in an elastic medium
If a source of oscillations is placed in any place of an elastic medium (solid, liquid, gaseous), then due to the interaction between particles, the oscillation will propagate in the medium from particle to hour

Equation of plane and spherical waves
The wave equation expresses the dependence of the displacement of an oscillating particle on its coordinates,

wave equation
The wave equation is a solution to a differential equation called the wave equation. To establish it, we find the second partial derivatives with respect to time and coordinates from the equation

12.4. Energy of a relativistic particle

12.4.1. Energy of a relativistic particle

The total energy of a relativistic particle is the sum of the rest energy of the relativistic particle and its kinetic energy:

E \u003d E 0 + T,

Equivalence of mass and energy(Einstein's formula) allows us to determine the rest energy of a relativistic particle and its total energy as follows:

• rest energy -

E 0 \u003d m 0 c 2,

where m 0 is the rest mass of a relativistic particle (the mass of the particle in its own frame of reference); c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s;

• total energy -

E \u003d mc 2,

where m is the mass of the moving particle (the mass of a particle moving relative to the observer with a relativistic velocity v); c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s.

Relationship between masses m 0 (mass of a particle at rest) and m (mass of a moving particle) is given by

Kinetic energy relativistic particle is determined by the difference:

T = E - E 0 ,

where E is the total energy of the moving particle, E = mc 2 ; E 0 - rest energy of the indicated particle, E 0 = m 0 c 2 ; the masses m 0 and m are related by the formula

m = m 0 1 − v 2 c 2 ,

where m 0 is the mass of the particle in the frame of reference relative to which the particle is at rest; m is the mass of the particle in the frame of reference relative to which the particle moves at a speed v; c is the speed of light in vacuum, c ≈ 3.0 ⋅ 10 8 m/s.

explicitly kinetic energy relativistic particle is defined by the formula

T = m c 2 − m 0 c 2 = m 0 c 2 (1 1 − v 2 c 2 − 1) .

Example 6. The speed of a relativistic particle is 80% of the speed of light. Determine how many times the total energy of the particle is greater than its kinetic energy.

Solution . The total energy of a relativistic particle is the sum of the rest energy of the relativistic particle and its kinetic energy:

E \u003d E 0 + T,

where E is the total energy of the moving particle; E 0 - rest energy of the specified particle; T is its kinetic energy.

It follows that the kinetic energy is the difference

T = E − E 0 .

The desired value is the ratio

E T = E E − E 0 .

To simplify the calculations, we find the reciprocal of the desired:

T E = E − E 0 E = 1 − E 0 E ,

where E 0 \u003d m 0 c 2; E = mc 2 ; m 0 - rest mass; m is the mass of the moving particle; c is the speed of light in vacuum.

Substituting the expressions for E 0 and E into the relation (T /E ) gives

T E = 1 − m 0 c 2 m c 2 = 1 − m 0 m .

The relationship between the masses m 0 and m is determined by the formula

m = m 0 1 − v 2 c 2 ,

where v is the speed of the relativistic particle, v = 0.80c.

Let's express the mass ratio from here:

m 0 m = 1 − v 2 c 2

and substitute it into (T /E ):

T E = 1 − 1 − v 2 c 2 .

Let's calculate:

T E \u003d 1 - 1 - (0.80 s) 2 c 2 \u003d 1 - 0.6 \u003d 0.4.

The desired value is the inverse ratio

E T \u003d 1 0.4 \u003d 2.5.

The total energy of a relativistic particle at the indicated speed exceeds its kinetic energy by a factor of 2.5.

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,