Comparison of the change in the potential energy of a stretched spring with the change in the kinetic energy of the body. When the position of the body changes, its potential energy changes. Thus Examples of problem solving

In the previous paragraph, it was found that when bodies interacting with each other by elastic force or gravity do work, then the relative position of the bodies or their parts changes. And when the work is done by a moving body, then its speed changes. But when the work is done, the energy of the bodies changes. Hence, we can conclude that the energy of bodies interacting by the force of elasticity or by the force of gravity depends on the relative position of these bodies or their parts. The energy of a moving body depends on its speed.

The energy of bodies, which they possess as a result of interaction with each other, is called potential energy. The energy of bodies, which they possess due to their movement, is called kinetic energy.

Consequently, the energy possessed by the Earth and the body located near it is the potential energy of the Earth-body system. For brevity, it is customary to say that this energy is possessed by the body itself, which is near the surface of the Earth.

The energy of a deformed spring is also potential energy. It is determined by the relative position of the spring coils.

Kinetic energy is the energy of motion. Kinetic energy can be possessed by a body that does not interact with other bodies.

Bodies can have both potential and kinetic energy at the same time. For example, an artificial satellite of the Earth has kinetic energy because it moves and potential energy because it interacts with the Earth by the force of gravity. The falling weight also has both kinetic and potential energy.

Now let's see how you can calculate the energy that a body has in a given state, and not just its change. For this purpose, it is necessary to select one specific state from various states of the body or system of bodies, with which all the others will be compared.

Let's call this state the "zero state". Then the energy of bodies in any state will be equal to the work that is done

upon transition from this state to the bullet state. (It is obvious that in the zero state the energy of the body is equal to a bullet.) Recall that the work done by the force of gravity and the force of elasticity does not depend on the trajectory of the body. It only depends on its starting and ending positions. In the same way, the work done when the speed of the body changes depends only on the initial and final speed of the body.

It makes no difference what state of bodies to choose as zero. But in some cases, the choice of the zero state suggests itself. For example, when it comes to the potential energy of an elastically deformed spring, it is natural to assume that the undeformed spring is in the zero state. The energy of an undeformed spring is zero. Then the potential energy of the deformed spring will be equal to the work that this spring would have done, passing into an undeformed state. When we are interested in the kinetic energy of a moving body, it is natural to take for zero the state of the body in which its velocity is equal to zero. We will get the kinetic energy of a moving body if we calculate the work that it would have done while moving to a complete stop.

It is a different matter when it comes to the potential energy of a body raised to a certain height above the Earth. This energy depends, of course, on the height of the body. But there is no "natural" choice of the zero state, that is, the position of the body from which its height should be measured. You can choose for zero the state of the body when it is on the floor of the room, at sea level, at the bottom of the mine, etc. It is only necessary when determining the energy of a body at different heights to count these heights from the same level, the height of which is assumed to be zero. Then the value of the body's potential energy at a given height will be equal to the work that would have been done when the body passed from this height to the zero level.

It turns out that, depending on the choice of the zero state, the energy of the same body has different meanings! This is no problem. Indeed, to calculate the work done by the body, we need to know the change in energy, that is, the difference between the two values ​​of energy. And this difference does not depend in any way on the choice of the zero level. For example, in order to determine how much the top of one mountain is higher than another, it makes no difference where the height of each peak is measured from. It is only important that it is measured from the same level (for example, from sea level).

The change in both kinetic and potential energy of bodies is always equal in absolute value to the work done by the forces acting on these bodies. But there is an important difference between both types of energy. The change kinetic energy the body, when a force acts on it, is really equal to the work done by this force, that is, it coincides with it both in absolute value and in sign. This follows directly from the theorem on

kinetic energy (see § 76). The change in the potential energy of bodies is equal to the work done by the forces of interaction, only in absolute value, and in sign it is opposite to it. Indeed, when the body, on which the force of gravity acts, moves downward, positive work is done, and the potential energy of the body decreases. The same applies to a deformed spring: when the stretched spring contracts, the elastic force does positive work, and the potential energy of the spring decreases. Recall that a change in a quantity is the difference between the subsequent and previous value of this quantity. Therefore, when a change in any quantity consists in the fact that it increases, this change has a positive sign. Conversely, if the value decreases, its change is negative.

Exercise # 54

1. In what cases does the body have potential energy?

2. In what cases does the body have kinetic energy?

3. What energy does a freely falling body have?

4. How does the potential energy of a body, which is acted upon by the force of gravity, change during its downward movement?

5. How will the potential energy of a body, which is acted upon by the force of elasticity or the force of gravity, change if, having passed along any trajectory, the body returns to its starting point?

6. How is the work done by the spring connected with a change in its potential energy?

7. How does the potential energy of a spring change when an undeformed spring is stretched? Are they squeezing?

8. The ball is suspended from the spring and vibrates. How does the potential energy of a spring change as it moves up and down?

There is a quite definite connection between the potential energy of a system of interacting bodies and the conservative force that determines the presence of this energy. Let's establish this connection.

1. If a conservative force acts on the body at every point in space, then they say that it is in potential field.

2. When the position of the body in this field changes, the potential energy of the body changes, while the conservative force does a very definite work. Let's express this work in the usual way.

We will assume that the body has moved in an arbitrary direction at an infinitely small distance
(fig. 25). Then

where
is the projection of the force vector onto the direction ... But
(19.2)

Equating the right-hand sides of expressions (19.1) and (19.2), we get:
, where
. (19.3)

is the derivative of the potential energy in the direction ; this value shows how fast the potential energy changes along this direction.

Thus, force projection to an arbitrary direction is equal in magnitude and opposite in sign derivative of potential energy in this direction.

Let's find out the meaning of the minus sign. If in the direction potential energy increases ( > 0), then according to (19.3) < 0. Это значит, что направление силыforms with direction obtuse angle, therefore, the component of this force acting along , opposite to the direction ... Conversely, if < 0, то проекция> 0, the angle between the force and direction spicy, co-

putting this force, acting along , coincides with the direction .

3. In the general case, the potential energy can change not only in the direction but also in any other direction. You can consider, for example, changes along the axes ,
Cartesian coordinate system.

Then
(19.4)

(icon means that it is taken private derivative).

Knowing the projection of force
it is easy to find the force vector:

. (19.5)

Taking into account (19.4) we will have:

. (19.6)

The vector on the right-hand side of relation (19.6) is called gradient magnitudes and denoted
.

Hence,

= -
. (19.7)

The conservative force acting on a body is equal in magnitude and opposite in direction to the gradient of the potential energy of this body. The potential energy gradient is a vector indicating the direction of the fastest increase in potential energy and is numerically equal to the change in energy per unit length of this direction.

When moving the body in direction actions of the conservative force occurs maximum work (since
= 1). But
... Therefore, the direction of force indicates the direction of the fastest reducing potential energy.

20 Graphic representation of potential

1. Potential energy is an coordinate function... In some simplest cases, it depends on only one coordinate (for example, in the case of a body raised above the Earth depends only on the height ). The dependence of the potential energy of the system on a particular coordinate can be represented graphically.

The graph depicting the dependence of the potential energy on the corresponding coordinate is called potential curve.

Let's analyze one of the possible potential curves (Fig. 26). Curve () shown in the figure shows how the potential energy of a system of particles changes if one of the particles moves along the axis , and all the rest remain in their places. Each point of the graph makes it possible to determine system corresponding to the particle coordinate .

2. The slope of the potential curve can be used to judge the magnitude and direction of the force acting on the particle along the corresponding directions. The magnitude and sign of the projection of this force on the considered direction is determined by the magnitude and sign of the tangent of the angle of inclination of the tangent to the curve at the appropriate points; in our case
, (20.1)

because
.

So than cooler there is a potential curve, so more force, acting on the particle along the corresponding direction. On the ascending sections of the potential curve, the tangents of the angles of inclination of the tangents are positive, therefore, the projection of the force negative. This means that the direction of the force acting along a given axis, opposite direction of this axis, the force prevents the removal of the particle from the system (Fig. 26, point ).

At the points corresponding to downward sections of the potential curve, the projection of the force positive, the force promotes the movement of the particle along a given direction (point ). At the points where
= 0, the force does not act on the particle (point ).

3. If, when one of the particles is removed (in any direction), the potential energy of the system sharply is increasing(the potential curve "soars" up), then they speak of the existence potential barrier. Talk about height barrier and its width in accordance with

SCH their places. So, if the particle is located at a point with a coordinate (Fig. 26), then its potential energy is
, the height of the potential barrier for it
, barrier width
... If a potential barrier is encountered on the path of a particle during its movement, both in the positive and negative directions of the chosen axis, then the particle is said to be in potential pit... The shape and depth of the potential well depends on the nature of the interaction forces and the configuration of the system.

4. Here are some examples. Figure 27 shows the potential

the real curve of a body raised above the Earth. As you know, the potential energy of such a body depends only on one coordinate - the height : = P.

Projection of gravity onto an axis is equal to
.

Z nak "minus" means that the direction of gravity is opposite to the direction of the axis ... Figure 28 shows the potential curve of a body fastened to a spring and oscillating. As can be seen from the figure, such a body is in a potential well with symmetrical walls. The potential energy of this body and the projection of the force acting on it are equal, respectively:

,
.

The curve shown in Fig. 29 is characteristic of the interaction of atoms and molecules in a solid. The peculiarity of this curve is that it is asymmetric; one edge of it is steep, the other is gentle.

Finally, the curve in Fig. 30 characterizes, in a first approximation, the potential energy of free electrons in the metal. The walls of this pit are almost vertical. This means that the force acting on the electrons at the metal boundary is very large.

G The smooth horizontal bottom of the well means that no force acts on the electrons inside the metal.

EXAMPLES OF SOLVING PROBLEMS

Example 1. Determine the work of compressing the spring of a railway car by 5 cm, if under the influence strength
the spring is compressed by

Solution. Neglecting the mass of the spring, we can assume that when it is compressed, only a variable pressure force acts, equal in magnitude to the elastic force determined by Hooke's law
... The work of this force when the spring is compressed by 5 cm must be determined. Counting on small displacement
force constant, we define elementary work as

.

Here the coefficient of spring stiffness is
.

We find the whole work by taking the integral of
ranging from NS 1 = 0 before

NS 2 = 5 cm.

After calculations, we will have

.

Example 2. Airplane mass m= 3 T must have a speed for takeoff =360km / h and takeoff run S=600 m. What is the minimum engine power required to take off the aircraft? Friction coefficient k wheels on the ground is 0.2. The aircraft acceleration motion is considered uniformly accelerated.

Solution. In the task it is required to determine instant motor power at the time of takeoff aircraft. It will be the minimum power at which the aircraft can still pick up the speed required for takeoff.

.

Traction force
we determine from the equation (second law of dynamics)

We find the acceleration from the equation of uniform motion
;

Taking into account the remarks made, the minimum power is

.

Example 3. The speed of a jet plane in a certain section varies with distance according to the law
... Find a job for a period of time (
if the mass of the aircraft m. At a moment in time speed is

Solution. Let us assume that the work is equal to the difference between the kinetic energies at the moments of time and , i.e.
... It is necessary to determine the law of speed variation with time. Airplane acceleration
Where
... After integrating and potentiating the last expression, we obtain that the speed at the moment of time is equal to

Thus, the work, for a given period of time, is equal to

Example 4. Body mass m under the influence of a constant wind force, it moves rectilinearly, and the dependence of the distance traveled on time changes according to the law
... Find the work of the wind force for the time interval from 0 to t.

Solution. The work of the wind force with a small displacement of the body is

, where we find the displacement as a derivative of the path with respect to time, i.e.
The force according to the second law of dynamics is

Full work for a period of time from 0 to t is equal to the integral of

Example 5. Ball mass
moves with speed
towards the ball with mass
moving with speed
... Find the value and explain the reason for the change in the kinetic energy of the system of balls after an inelastic central impact.

Solution. Ball system energy before impact

After an inelastic impact, the balls will move at the same speed. u, which we find by applying the law of conservation of momentum

Energy of the system of balls after impact

.

Decrease in kinetic energy after impact

The change in kinetic energy is spent on deformation and, ultimately, on heating the balls:

Example 6. Vehicle weight
moving along the horizontal section of the track at a speed
, develops a power equal to
... What power should the car develop when driving it uphill with a slope
with the same speed?

Determine the steepness of the descent (angle of inclination) along which the car will go at a speed of 30 km / h, with the engine off.

Solution. 1) The power of the car when driving uphill will be determined by the traction force and the speed of movement

The friction force is defined as
, where the force of normal pressure on an inclined plane
... If we consider the coefficient of friction to be the same along the entire path of movement, then on the horizontal section it is equal to
... The friction force can be found from the ratio (with uniform horizontal movement)
, i.e.
and
... Then the friction force on an inclined plane

The shearing force is
... Taking into account the comments made, the power of a car moving uphill will be equal to

Let's substitute these tasks

2) When driving downhill with the engine off, the traction force is zero. Only the rolling force is active
and friction force
Given their direction

-
,

where

.

Thus, the slope is equal to
.

Example 7. A heavy ball slides without friction along an inclined groove that forms a “dead loop” of radius R... From what height should the ball begin to move in order not to break away from the chute at the top of the trajectory?

Solution. The problem of non-uniformly variable motion of a material point along a circle is given. Moreover, in the process of movement, the position of the body in height changes. Such problems are solved using the law of conservation of energy and drawing up an equation according to the second law of dynamics for the direction of the normal. Since the energy remains unchanged for a closed system, we will write it in the form
.

Let's take the beginning of the movement as the initial position of the ball, and the position at the top point of the trajectory as the final one. Set the height reference level from the table surface.

Ball energy in first position
, in the second position
... Hence
, where

. (1)

For determining h it is necessary to know the speed of the ball at the top point. In this case, we will take into account that at the top point of the loop on the ball, in the general case, two forces act downward - the force of gravity R and the reaction force from the support side N. Under the action of these forces, the ball moves in a circle, i.e.

When descending from a sufficiently high altitude, the ball acquires such speed that at each point of the loop it presses on the chute with some force ... According to Newton's third law, the groove acts on the ball with the same force N in the opposite direction and squeezes it to the arc of a circle of radius R.

As the initial height decreases, the speed of the ball decreases and at a certain value h becomes such that it flies past the top point of the loop, only touching the groove. For such a limiting case N = 0 and the equation of the second law of dynamics takes the form

or

where
(2)

Substituting (2) into (1) and solving the last equation for h, we get

QUESTIONS FOR SELF-TEST.

1. What is called energy? What is called kinetic energy? What is called potential energy?

2. What is work? How is the work of constant and variable force calculated?

3. What is power?

4. What is the relationship between mechanical work and kinetic energy?

5. Prove that gravity is a conservative force.

6. What is the relationship between the work of conservative forces and potential energy?

7. What is zero potential energy? How is he chosen?

8. What is the relationship between the potential energy of the body and the conservative force acting on it?

9. What is a potential hole and a potential barrier?

USED ​​BOOKS

IV Saveliev General physics course: in 3 volumes; textbook for universities. Vol. 1: Mechanics. Molecular physics. / I.V. Savelyev.-4th ed. ster. -SPb .: Lan, 2005.

Zisman G.A., Course of General Physics. T.1 / G.A. Zisman, O.M. Todes.– M.: Nauka, 1972.

Detlaf A.A.Physics course: tutorial for technical colleges. / A.A. Detlaf, B.M. Yavorsky.-4th ed., Rev. - M .: Higher school, 2002. - 718 p.

Trofimova T.I. Physics course: textbook for universities. / T.I. Trofimova. - 7th ed., Sr. - M .: Higher. shk., 2001.- 541 p.

A.G. Chertov Problem book on physics: a textbook for technical colleges. / A.G. Chertov, A.A. Vorobyov. - 8th ed., Revised. and additional, Moscow: Fizmatlit, 2006, 640 p.

1. You got acquainted with the concept of energy in the 7th grade physics course. Let's remember him. Suppose that some body, for example a trolley, slides off an inclined plane and moves a bar lying at its base. They say that the cart does the job. Indeed, it acts on the bar with a certain elastic force and the bar moves at the same time.

Another example. The driver of a car moving at a certain speed applies the brake, and the car stops after a while. In this case, too, the car does work against the frictional force.

They say that if the body can do work, then it has energy.

Energy is denoted by the letter E... The unit of energy in SI is joule (1 J).

2. There are two types of mechanical energy - potential and kinetic.

Potential energy is the energy of interaction of bodies or body parts, depending on their relative position.

All interacting bodies possess potential energy. So, any body interacts with the Earth, therefore, the body and the Earth have potential energy. The particles that make up the bodies also interact with each other, and they also have potential energy.

Since potential energy is the energy of interaction, it refers not to one body, but to a system of interacting bodies. In the case when we are talking about the potential energy of a body raised above the Earth, the system is composed of the Earth and the body raised above it.

3. Let us find out what the potential energy of a body raised above the Earth is equal to. To do this, we will find a connection between the work of the force of gravity and the change in the potential energy of the body.

Let the body mass m falls from a height h 1 up to height h 2 (fig. 72). In this case, the displacement of the body is h = h 1 – h 2. The work of gravity in this area will be equal to:

A = F heavy h = mgh = mg(h 1 – h 2), or
A = mgh 1 – mgh 2 .

The quantity mgh 1 = E n1 characterizes the initial position of the body and represents its potential energy in the initial position, mgh 2 = E n2 - potential energy of the body in the final position. The formula can be rewritten as follows:

A = E n1 - E n2 = - ( E n2 - E n1).

When the position of the body changes, its potential energy changes. Thus,

the work of gravity is equal to the change in the potential energy of the body, taken with the opposite sign.

The minus sign means that when the body falls, the force of gravity does positive work, and the potential energy of the body decreases. If the body moves up, then the force of gravity does negative work, and the potential energy of the body increases.

4. When determining the potential energy of a body, it is necessary to indicate the level relative to which it is measured, called zero level.

So, the potential energy of a ball flying over a volleyball net has one meaning with respect to the net and another with respect to the floor of the gym. It is important here that the difference between the potential energies of the body at two points does not depend on the selected zero level. This means that the work done by the potential energy of the body does not depend on the choice of the zero level.

Often the surface of the Earth is taken as the zero level when determining the potential energy. If a body falls from a certain height to the surface of the Earth, then the work of gravity is equal to potential energy: A = mgh.

Hence, the potential energy of a body raised to a certain height above the zero level is equal to the work of the force of gravity when the body falls from this height to the zero level.

5. Any deformed body possesses potential energy. When a body is compressed or stretched, it is deformed, the forces of interaction between its particles change, and an elastic force arises.

Let the right end of the spring (see Fig. 68) move from a point with coordinate D l 1 to the point with coordinate D l 2. Recall that the work of the elastic force is equal to:

A =– .

Quantity = E n1 characterizes the first state of the deformed body and represents its potential energy in the first state, the value = E n2 characterizes the second state of the deformed body and represents its potential energy in the second state. You can write:

A = –(E n2 - E n1), i.e.

the work of the elastic force is equal to the change in the potential energy of the spring, taken with the opposite sign.

The minus sign shows that as a result of positive work, perfect force of elasticity, the potential energy of the body decreases. When a body is compressed or stretched under the action of an external force, its potential energy increases, and the elastic force performs negative work.

Self-test questions

1. When can you say that the body has energy? What is the unit of energy?

2. What energy is called potential?

3. How to calculate the potential energy of a body raised above the Earth?

4. Does the potential energy of a body raised above the Earth depend on the zero level?

5. How to calculate the potential energy of an elastically deformed body?

Task 19

1. What work must be done to transfer a bag of flour weighing 2 kg from a shelf located at a height of 0.5 m relative to the floor to a table located at a height of 0.75 m relative to the floor? What is the potential energy of a bag of flour lying on the shelf with respect to the floor, and its potential energy when it is on the table?

2. What work must be done to transfer a spring with a stiffness of 4 kN / m to the state 1 stretching it 2 cm? What additional work must be done to transfer the spring to the state 2 by stretching it another 1 cm? What is the change in the potential energy of the spring when it is transferred to the state 1 and from the state 1 in a state 2 ? What is the potential energy of the spring in the state 1 and able to 2 ?

3. Figure 73 shows a graph of the dependence of the force of gravity acting on the ball on the height of the ball. Calculate the potential energy of the ball at a height of 1.5 m using the graph.

4. Figure 74 shows a graph of the dependence of the elongation of the spring on the force acting on it. What is the potential energy of the spring with an elongation of 4 cm?

Kinetic energy is the energy of a mechanical system, depending on the speed of movement of its points in the selected frame of reference. The kinetic energy of translational and rotational motion is often isolated. In simple terms, kinetic energy is the energy that a body only has when it moves. When the body is not moving, kinetic energy is zero. Work and body speed change. Let us establish a connection between the work of a constant force and a change in the speed of the body. In this case, the work of force can be defined as. The modulus of force according to Newton's second law is equal, and the modulus of displacement for uniformly accelerated rectilinear motion

. (19.3) The work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body. This statement is called the kinetic energy theorem.

Since the change in kinetic energy is equal to the work of the force (19.3), the kinetic energy is expressed in the same units as the work, i.e. in joules.

If the initial speed of movement of a body with a mass is equal to zero and the body increases its speed to a value, then the work of the force is equal to the final value of the kinetic energy of the body:

... (19.4) Since the displacement coincides in direction with the vector of gravity, the work of gravity is

... (20.1) that the work of the force of gravity does not depend on the trajectory of the body and is always equal to the product of the modulus of the force of gravity by the difference in heights in the initial and final positions. When moving down, the work of gravity is positive; when moving up, it is negative. The work of gravity on a closed path is zero. The value of the potential energy of a body raised above the Earth depends on the choice of the zero level, i.e. the height at which the potential energy is taken to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.

Solutions, osmotic pressure. Humidity: relative and absolute humidity, dew point. Osmotic pressure(denoted by π) - excess hydrostatic pressure on the solution, separated from the pure solvent by a semipermeable membrane, at which the diffusion of the solvent through the membrane (osmosis) stops. This pressure tends to equalize the concentrations of both solutions due to the counter-diffusion of solute and solvent molecules. The magnitude of the osmotic pressure created by the solution depends on the amount, and not on the chemical nature of the substances dissolved in it (or ions, if the molecules of the substance dissociate), therefore, the osmotic pressure is a colligative property of the solution.

The higher the concentration of a substance in a solution, the greater the osmotic pressure created by it. This rule, which is called the law of osmotic pressure, is expressed by a simple formula, very similar to a certain law of an ideal gas:, where i is the isotonic coefficient of the solution; C is the molar concentration of the solution, expressed through a combination of basic SI units, that is, in mol / m 3, and not in the usual mol / l; R is the universal gas constant; T is the thermodynamic temperature of the solution.

Absolute air humidity (f) is the amount of water vapor actually contained in 1m 3 of air: f = m (mass of water vapor contained in the air) / V (volume of humid air). The commonly used unit for absolute humidity is: (f) = g / Relative humidity: φ = (absolute humidity) / (maximum humidity). Relative humidity is usually expressed as a percentage. These values ​​are related to each other by the following ratio: φ = (f × 100) / fmax. The dew point is the temperature to which the air must cool in order for the vapor contained in it to reach saturation and begin to condense into dew.

Crystalline and amorphous solids. Liquid crystals. Deformation of solids. Types of deformation.

Solid- the aggregate state of matter, characterized by the constancy of the form and the nature of the movement of atoms, which perform small vibrations around the equilibrium positions. Crystalline bodies... A solid body is difficult to compress or stretch under normal conditions. To give solids the desired shape or volume in factories and factories, they are processed on special machines: turning, planing, grinding. Amorphous bodies... In addition to crystalline, amorphous bodies are also referred to as solids.

AT- this is solids, which are characterized by a disordered arrangement of particles in space. Amorphous bodies include glass, amber, various other resins, and plastics. Although at room temperature these bodies retain their shape, as the temperature rises, they gradually soften and begin to flow like liquids: amorphous bodies do not have a certain temperature or melting. Liquid crystals - This is a phase state, into which some substances pass under certain conditions (temperature, pressure, concentration in solution).

LCD possess simultaneously the properties of both liquids (fluidity) and crystals (anisotropy). Deformation of a solid- change in the linear dimensions or shapes of a solid under the influence of external forces. Types of deformations : Deformation stretching or compression- change of any linear size of the body (length, width or height). Deformation shift- displacement of all layers of a solid in one direction parallel to a certain shear plane. Deformation bending- compression of some parts of the body while stretching others. Deformation torsion- rotation of parallel sections of the sample around a certain axis under the action of an external force.

Mechanical properties of solids. Hooke's Law. Deformation curve. Limits of elasticity and strength. Plastic deformation.

Under the action of applied external forces, rigid bodies change their shape and volume - they are deformed. If, after the cessation of the action of the force, the shape and volume of the body are completely restored, then the deformation is called elastic, and the body is absolutely elastic. Deformations that do not disappear after the cessation of the action of the forces are called plastic and the bodies are plastic. There are the following types of deformations: tension, compression, shear, torsion and bending. Tensile deformation is characterized by absolute elongation delta l and relative elongation e: where l 0- initial length, l is the final length of the bar. Mechanical stress is the ratio of the modulus of the elastic force F to the cross-sectional area of ​​the body S: b = F / S.

In SI, 1Pa = 1N / m 2 is taken as a unit of mechanical stress. Hooke's Law: at small deformations, the stress is directly proportional to the relative elongation (b= E. e). Elastic deformation is called one in which, after the cessation of the action of the force, the body restores its original shape and size. Plastic deformation name. such that, after the termination of the load, the body does not restore its original shape and size. Plastic deformation is always preceded by elastic deformation.

The basic equation of the molecular kinetic theory of gases.

The ideal gas model is used to explain the properties of a substance in a gaseous state. The ideal gas model assumes the following: molecules have a negligible volume compared to the volume of the vessel, attraction forces do not act between molecules, and repulsive forces act when molecules collide with each other and with the vessel walls. Ideal gas pressure. One of the first and important successes of the molecular kinetic theory was a qualitative and quantitative explanation of the phenomenon of gas pressure on the walls of a vessel. A qualitative explanation of the pressure with the walls of the vessel interact with them according to the laws of mechanics as elastic bodies... When a molecule collides with a vessel wall, the projection of the velocity vector onto the OX axis, perpendicular to the wall, changes its sign to the opposite, but remains constant in absolute value

Therefore, as a result of the collision of the molecule with the wall, the projection of its momentum on the OX axis changes from to. A change in the momentum of a molecule shows that a force directed from the wall acts on it in a collision. The change in the momentum of the molecule is equal to the momentum of the force: During the collision, the molecule acts on the wall with a force equal to the force in modulus according to Newton's third law and directed oppositely. There are a lot of gas molecules, and their impacts on the wall follow one after the other with a very high frequency. The average value of the geometric sum of forces acting on the part of individual molecules when they collide with the vessel wall is the force of gas pressure. The gas pressure is equal to the ratio of the modulus of the pressure force to the wall area S: Based on the use of the basic provisions of the molecular kinetic theory, an equation was obtained that made it possible to calculate the gas pressure if the mass m0 of the gas molecule, the mean value of the square of the velocity of the molecules and the concentration of n molecules are known: the equation is called the basic equation of molecular kinetic theory. Having denoted the average value of the kinetic energy of the translational motion of the ideal gas molecules: we get. The pressure of an ideal gas is equal to two-thirds of the average kinetic energy of the translational motion of the molecules contained in a unit volume.

Internal energy of the system as a function of state. Equivalence of heat and work. The first law of thermodynamics.

Internal energy - thermodynamic function the state of the system, its energy, determined by the internal state. It consists mainly of the kinetic energy of motion of particles (atoms, molecules, ions , electrons) and the energy of interaction between them (intra- and intermolecular). The internal energy is affected by a change in the internal state of the system under the influence of an external field; the internal energy includes, in particular, the energy associated with the polarization of the dielectric in an external electric field and magnetization of the paramagnet in an external magnetic field.

The kinetic energy of the system as a whole and the potential energy due to the spatial arrangement of the system are not included in the internal energy. In thermodynamics, only the change in internal energy in various processes is determined. Therefore, the internal energy is specified up to a certain constant term, depending on the energy taken as zero. Internal energy U as a function of state is introduced by the first law of thermodynamics, according to which the difference between the heat Q transferred to the system and the work W performed by the system depends only on the initial and final states of the system and does not depend on the transition path, i.e. represents the change in the state function Δ U where U 1 and U 2- internal energy of the system in the initial and final states, respectively. Equation (1) expresses the law of conservation of energy as applied to thermodynamic processes, i.e. processes in which heat transfer occurs. For a cyclic process that returns the system to its initial state, Δ U= 0. In isochoric processes, i.e. processes at a constant volume, the system does not perform work due to expansion, W= 0 and the heat transferred to the system is equal to the increment of internal energy: Q v= Δ U... For adiabatic processes, when Q= 0, Δ U= -W. Internal energy system as a function of its entropy S, volume V and the number of moles m i of the i-th component is the thermodynamic potential. This is a consequence of the first and second principles of thermodynamics and is expressed by the ratio:

Relative dielectric constant. Electric constant. Electric field strength.

The dielectric constant environment - a physical quantity characterizing the properties of an insulating (dielectric) medium and showing the dependence of electrical induction on the strength electric field... The relative permittivity ε is dimensionless and shows how many times the force of interaction of two electric charges in a medium is less than in a vacuum. This value for air and most other gases in normal conditions close to unity (due to their low density).

For most solid or liquid dielectrics, the relative permittivity ranges from 2 to 8 (for a static field). The dielectric constant of water in a static field is quite high - about 80. Electric constant (e 0) - physical constant included in the ur-tion of the laws of electric. fields (e.g. in Pendant law) when writing these ur-ny in a rationalized form, in accordance with a cut formed electric. and magn. units International System of Units; according to the old terminology, an e. item is called dielectric. permeability of vacuum. where m 0 - magnetic constant. Unlike dielectric. permeability e, depending on the type of substance, temperature, pressure and other parameters, E. p. e 0 depends only on the choice of the system of units.

For example, in Gaussian CGS system of units electric field strength in classical electrodynamics ( E) is the vector characteristic of electric. field, the force acting on the unit at rest in the given frame of reference. charge. In this case, it is assumed that the introduction of a charge (charged test body) in the external. field E does not change that. Sometimes, instead of H. e. etc. they simply say "electric field". Dimension N. e. p. in the Gaussian system - L -1/2 M 1/2 T -1, in SI - LMT -3 I -1; unit H. e. p. in SI is volt per meter (1 CGSE = 3.10 4 V / m). Distribution of H. e. n. in space is usually characterized by a family of lines E(the lines of force of the electric field) tangent to the to-psh at each point coincide with the directions of the vector E.

Like any vector field, the field E is divided into two components: potential ([ E n) = 0, E n = - j e) and vortex ( E B = 0, E B = [ A m]). In particular, electric. the field created by the system of stationary charges is purely potential. Electric. radiation field, including field E in transverse e-mag. waves, is purely vortex. Together with the vector magn. induction V H. e. n. constitutes a single 4-tensor of the electromagnetic field.

Therefore, purely electric. the field of a given system of charges exists only in the "chosen" frame of reference, where the charges are motionless. In other inertial frames of reference, moving relative to the "chosen" with a post. speed u, there is also a magnetic field V" = = [uE] / due to the appearance of convection. currents j= r u/ (r is the charge density in the "chosen" system).

Energy is a scalar quantity. In the SI system, the unit of measure for energy is Joule.

Kinetic and potential energy

There are two types of energy - kinetic and potential.

DEFINITION

Kinetic energy Is the energy that the body possesses due to its movement:

DEFINITION

Potential energy- This is energy, which is determined by the mutual arrangement of bodies, as well as the nature of the forces of interaction between these bodies.

The potential energy in the gravitational field of the Earth is the energy due to the gravitational interaction of the body with the Earth. It is determined by the position of the body relative to the Earth and is equal to the work of moving the body from a given position to the zero level:

Potential energy - energy due to the interaction of body parts with each other. It is equal to the work of external forces in tension (compression) of an undeformed spring by the value:

The body can simultaneously possess both kinetic and potential energy.

Full mechanical energy a body or a system of bodies is equal to the sum of the kinetic and potential energies of the body (system of bodies):

Law of energy conservation

For a closed system of bodies, the law of conservation of energy is valid:

In the case when external forces act on a body (or a system of bodies), for example, the law of conservation of mechanical energy is not fulfilled. In this case, the change in the total mechanical energy of the body (system of bodies) is equal to external forces:

The law of conservation of energy makes it possible to establish a quantitative relationship between various forms of motion of matter. As well as, it is valid not only for, but also for all natural phenomena. The law of conservation of energy says that energy in nature cannot be destroyed in the same way as it can be created from nothing.

In the most general view the energy conservation law can be formulated as follows:

• energy in nature does not disappear and is not created again, but only transforms from one type to another.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2