Sustainable and unstable equilibrium of a solid. T. Types of equilibrium

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Objectives lesson:Study the balance of equilibrium bodies, to get acquainted with various types of equilibrium; Find out the conditions in which the body is in equilibrium.

Tasks lesson:

Training:Examine two equilibrium conditions, equilibrium types (sustainable, unstable, indifferent). To find out under what conditions the body is more stable.
Developing:Promote the development of cognitive interest in physics. Development of skills to compare, generalize, allocate the main thing, draw conclusions.
Educational:Educating attention, the ability to express your point of view and defend it, develop the communicative abilities of students.

Type of lesson:lesson studying new material with computer support.

Equipment:

Disk "Work and Power" from "Electronic lessons and tests.
Table "equilibrium conditions".
Prism leaving with a plumb.
Geometric bodies: cylinder, cube, cone, etc.
Computer, Multimediaprocket, Interactive Board or Screen.
Presentation.

During the classes

Today, at the lesson, we will find out why the lifting crane does not fall, why the toy "Vanka-stand" always returns to its original state, why does not the Pisa Tower fall?

I. Repetition and actualization of knowledge.

Formulate the first Newton law. What condition is stated in the law?
What question answers the second law Newton? Formula and wording.
What question is the third law of Newton? Formula and wording.
What is called the resultant force? How is it located?
From the disk "Movement and interaction of the bodies", to complete the Task number 9 "Ensucting forces with different directions" (vector addition of vectors (2, 3 exercises)).

II. Studying a new material.

1. What is called equilibrium?

Equilibrium is a state of rest.

2. Equilibrium conditions.(Slide 2)

a) When is the body alone? What law is it follows?

First equilibrium condition:The body is in equilibrium, if the geometric sum of the external forces applied to the body is zero. ΣF \u003d 0.

b) Let two equal power act on the board as shown in the figure.

Will she be in equilibrium? (No, she will turn)

Only the center point is located alone, and the rest move. It means that the body is in equilibrium, it is necessary that the sum of all forces acting on each element is 0.

The second equilibrium condition: The sum of the moments of the forces acting clockwise should be equal to the sum of the moments of the forces acting counterclockwise.

Σ m clockwise \u003d Σ m counterclockwise

Power moment: M \u003d F L

L - shoulder strength - the shortest distance from the point of support to the line of action.

3. The center of gravity of the body and its finding.(Slide 4)

Center of gravity body - This is a point through which the resultant of all parallel gravity forces acting on individual elements of the body (with any position of the body in space).

Find the center of gravity of the following figures:

4. Types of equilibrium.

but) (slides 5-8)

Output: Equilibrium is steadily, if with a small deviation from the equilibrium position there is a force seeking to return it to this position.

A steadily position in which its potential energy is minimal. (Slide 9)

b) the stability of the bodies located at the point of the support or on the support line. (slides 10-17)

Output:For the resistance of the body located at one point or the support line, it is necessary that the center of gravity is below the plot point (line).

c) the stability of bodies on a flat surface.

(Slide 18)

1) Surface support - This is not always a surface that comes into contact with the body (and the one, which is limited to the lines connecting the feet of the table, tripods)

2) Disselection of the slide from "Electronic lessons and tests", disk "Work and power", lesson "Types of equilibrium".

Picture 1.

What differ stools? (Support Square)
Which one is more stable? (With a larger area)
What differ stools? (Location of the center of gravity)
Which one is most stable? (Choosing the center of gravity below)
Why? (Because it can be rejected to a larger angle without tipping)

3) Experience with priscious deviating

We put a prism with a plumb on the board and begin to gradually raise it for one edge. What do we see?
While the plumb line crosses the surface bounded by the support, the balance is preserved. But as soon as the vertical passing through the center of gravity, will begin to go beyond the boundaries of the surface of the support, the shelf rolls over.

Parse slides 19-22..

Conclusions:

Sustainably, the body, which has more support area.
Of the two bodies of the same area, the body that the center of gravity is below, because It can be rejected without tipping to a large angle.

Parse slides 23-25.

What ships are most stable? Why? (Whose cargo is located in the holds, and not on the deck)

What cars are most stable? Why? (To increase the stability of machines on turns, the canvas road til down in the direction of rotation.)

Conclusions:Equilibrium can be stable, unstable, indifferent. The stability of the bodies is the greater the greater the area of \u200b\u200bthe support and below the center of gravity.

III. Application of knowledge of stability tel.

What specialties are the most necessary knowledge about equilibrium bodies?
Designers and designers of various structures (high-rise buildings, bridges, television towers, etc.)
Circus artists.
Drivers and other specialists.

(slides 28-30)

Why is the "Vanka-stand" returns to the equilibrium position in any slope of the toy?
Why is the Pisa Tower standing under the tilt and does not fall?
How do equilibrium cyclists and motorcyclists retain?

Conclusions from the lesson:

There are three types of equilibrium: sustainable, unstable, indifferent.
Stable body position in which its potential energy is minimal.
The stability of bodies on a flat surface is the greater, the greater the support area and below the center of gravity.

Homework: § 54. – 56 (G.Ya. Myakyshev, B.B. Bukhovtsev, N.N. Sotsky)

Used sources and literature:

G.Ya. Myakyshev, B.B. Bukhovtsev, N.N.Sotsky. Physics. Grade 10.
Filter "Stability" of 1976 (scanned by me on the film scanner).
Disk "Movement and interaction of bodies" from "Electronic lessons and tests".
Disk "Work and Power" from "Electronic Lessons and Tests".

Market equilibrium is called stable if there is a market forces restoring it from an equilibrium state. Otherwise, equilibrium is unstable.

To check whether the situation presented in Fig. 4.7, sustainable equilibrium, assume that the price has risen from R0 BE P.1. As a result, an excess in the amount Q2 - Q1 is formed on the market. Regarding what will happen after this, there are two versions: L. Valras and A. Marshall.

According to L. Valras, in excess there is a competition between sellers. To attract buyers, they will begin to reduce the price. As price decreases, the volume of demand will increase, and the amount of proposal is reduced until the initial equilibrium is restored. In case of rejecting the price down from its equilibrium value, the demand will exceed the offer. Competition will begin between buyers

Fig. 4.7.Restoration of equilibrium. Pressure: 1 - by Marshall; 2 - Valrasu

for scarce goods. They will offer sellers a higher price, which will increase the offer. So it will continue until the price is returned to the equilibrium level P0. Therefore, Valrasu combination P0, Q0 represents a steady market equilibrium.

Although A. Marshall reasoned. When the amount of supply is less than equilibrium, then the price of demand exceeds the offer price. Firms receive a profit that stimulates the expansion of production, and the supply will increase until it reaches the equilibrium value. In case of exceeding the equilibrium volume of supply, the demand price will be lower than the price of the offer. In such a situation, entrepreneurs suffer losses, which will lead to a decrease in production to an equilibrium break-even. Consequently, in the Marshall, the intersection point of the curves of supply and supply in Fig. 4.7 Represents a steady market balance.

According to L. Valras, buyers are in terms of the active side of the market, and in excess sellers. According to A. Marshall, the dominant force in the formation of market conditions is always entrepreneurs.

However, the two considered variations of the diagnosis of the stability of the market equilibrium lead to the same result only in cases of a positive inclination of the curve of supply and negative - the demand curve. When it is not, then the diagnosis of the stability of equilibrium states of the market for Valras and Marshall does not coincide. Four versions of such states are shown in Fig. 4.8.

Fig. 4.8.

Situations presented in Fig. 4.8, and in, Possible under the growing effect of the scale when manufacturers can reduce the price of the offer as the release increases. The positive inclination of the demand curve in situations shown in Fig. 4.8, b, g, can reflect the paradox of the hyffen or the snob effect.

Valrasu sectoral equilibrium presented in Fig. 4.8, a, b, is unstable. If the price rises to R1, the market will arise a deficit: QD\u003e QS. In such conditions, buyers competition will cause further price increase. If the price drops to P0, then the offer will exceed the demand that the Valrass should lead to a further decrease in the price. Marshall combination P *, Q * Represents a steady balance. With less than Q *, the offer price of demand will be higher than the price of the offer, and this stimulates an increase in release. In case of increasing Q * the price of demand will be below the price of the offer, so it will decrease.

When the demand curves and supply are located as shown in Fig. 4.8, in, g, Then, by the logic of Valras, balance at the point P *, Q * Stable, since at P1\u003e P * an excess occurs, and when P0< Р* –дефицит. По логике Маршалла–это варианты неустойчивого равновесия, так как при Q < Q* цена предложения оказывается выше цены спроса, предложение будет уменьшаться, а в случае Q > Q * - On the contrary.

The discrepancies between L. Valrasov and A. Marshall in describing the mechanism of the market functioning are caused by the fact that, in the opinion of the first, market prices are completely flexible and instantly react to any changes in the situation, and in the opinion of the second, prices are not flexible enough and in the event of disproportions between demand and By proposal, market transactions react faster on them than prices. The interpretation of the process of establishing a market equilibrium for Valras complies with the conditions of perfect competition, and in the Marshall - imperfect competition in the short period.

L. Valras (1834-1910) is the founder of the concept of general economic equilibrium.

Static is the section of mechanics that study the conditions of equilibrium tel.

From the second law of Newton, it follows that if the geometric sum of all external forces applied to the body is zero, then the body is at rest or performs a uniform straight line. In this case, it is customary to say that the forces attached to the body, balassize each other. When calculating directifying All the forces acting on the body can be applied to center Mass. .

So that the infrared body is in equilibrium, it is necessary to be equal to zero that is equal to all forces applied to the body.

In fig. 1.14.1 This is an example of a solid body equilibrium under the action of three forces. Intersection point O. The lines of force and does not coincide with the point of application of gravity (center of mass C.), But with equilibrium, these points are necessarily located on one vertical. When calculating the resultant all forces are given to one point.

If the body can rotate relative to some axis, then for its equilibrium there is not enough equality to zero however, all forces.

The torque effect depends not only on its size, but also from the distance between the force line and the axis of rotation.

The length of the perpendicular spent from the axis of rotation to the line of action is called shoulder power.

Module Module on Shoulder d. called moment of power M.. Positive are the moments of those forces that seek to turn the body counterclockwise (Fig. 1.14.2).

Moment rule : The body having a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all the forces attached to the body relative to this axis are zero:

In the international system of units (s) moments of forces are measured in N.tuteton- meters (N ∙ M.) .

In general, when the body can move progressively and rotate, for equilibrium, it is necessary to perform both conditions for equilibrium: equality zero of the resultant force and equality zero the sum of all moments of forces.

here is a screenshot of the game about equilibrium

Riding horizontal surface wheel - example indifferent equilibrium (Fig. 1.14.3). If you stop the wheel anywhere, it will be in an equilibrium state. Along with indifferent equilibrium in mechanics distinguish states sustainable and unstable Equilibrium.

The state of equilibrium is called stable if, with small deviations of the body, the forces or moments of forces occur from this state, seeking to return the body into an equilibrium state.

With a small deviation of the body from the state of unstable equilibrium, there are forces or moments of forces, seeking to remove the body from the equilibrium position.

The ball lying on a flat horizontal surface is in a state of indifferent equilibrium. The ball located at the upper point of the spherical protrusion is an example of an unstable equilibrium. Finally, the ball at the bottom of spherical recess is in a state of stable equilibrium (Fig. 1.14.4).

For a body having a fixed axis of rotation, all three types of equilibrium are possible. Indeline equilibrium occurs when the axis of rotation passes through the center of mass. With a stable and unstable equilibrium, the center of mass is located on a vertical direct passing through the axis of rotation. At the same time, if the center of the masses is below the axis of rotation, the equilibrium state turns out to be stable. If the center of the masses is above the axis - the equilibrium state is unstable (Fig. 1.14.5).

A special case is the balance of the body on the support. In this case, the elastic strength of the support is applied not to one point, but is distributed over the base of the body. The body is in equilibrium if the vertical line conducted through the center of mass body passes through square support, i.e., inside the contour formed by the lines connecting the points of the support. If this line does not intersect the support area, then the body overturns. An interesting example of an equilibrium of the body on the support is the falling tower in the Italian city of Pisa (Fig. 1.14.6), which, according to legend, used Galilee when studying the laws of free falling tel. The tower has a form of a cylinder with a height of 55 m and a radius of 7 m. The top of the tower is rejected from the vertical of 4.5 m.

The vertical line conducted through the center of the mass of the tower crosses the base of approximately 2.3 m from its center. Thus, the tower is in a state of equilibrium. Equilibrium will break and the tower will fall when the deviation of its vertical from the vertical will reach 14 m. Apparently, it will happen very soon.

Page 1.

Unstable equilibrium is characterized by the fact that the system, being removed from equilibrium, does not return to the initial state, and goes into another sustainable state. Systems can be in a state of unstable equilibrium for a short period of time. In practice, there are semi-resistant (metastable) states resistant to a more remote state. Metastable states are possible in cases where the characteristic functions have several extremum points. After some time expires, the system in a metastable state passes into a stable (stable) state.

Unstable equilibrium differs from stable in that the system, being derived from the equilibrium state, is not returned to the initial state, but enters a new steady balance of equilibrium.

An unstable balance occurs when some deviation from equilibrium prices creates strength, seeking to move prices further and further from the state of equilibrium. In the analysis of supply and demand, such a phenomenon may occur when both curves are supply and demand - have a negative slope and the proposal curve crosses the demand curve from above. If it crosses it from the bottom, then the steady equilibrium still comes. The status of equilibrium may not occur at all. Using an example with the curves of supply and demand, it is possible to show that there are cases in which curves do not intersect, and, therefore, there is no equilibrium price, as there is no price that would arrange and buyers and sellers. And the last - the demand curves and suggestions can cross more than once, and then there may be several equilibrium prices, and there will be a steady equilibrium with each of them.

Unstable equilibrium is characterized by the fact that the body deflected from the initial position does not return to it and does not remain in the new position. And, finally, if the body remains in a new position and does not seek to return to the initial, then the balance is called indifferent.

Unstable equilibrium differs from sustainable in that the system, being removed from the state of equilibrium, is not refundable to the initial state, but enters a new, stable balance of equilibrium.

Unstable equilibrium differs from sustainable in that the system, being removed from the state (equilibrium, is not returned to the initial state, but it goes into a new - stable state of equilibrium.

Unstable balance, if the body, being derived from the equilibrium position in the nearby situation and then provided to himself, will further deviate from this situation.

Unstable equilibrium occurs if the body, being derived from the position of the equilibrium in the near position and then provided to himself, will further deviate from this equilibrium situation.

Unstable equilibrium differs from sustainable in that the system, being derived from equilibrium state, is not returned to the initial state, but it moves into a new and moreover a stable state of equilibrium. Unstable equilibrium cannot exist and therefore is not considered in thermodynamics.

Unstable equilibrium is practically impossible, since it is impossible to isolate the system from infinitely small external influences.

An unstable balance between supply and oil supply and prospects for ensuring a smooth transition by achieving the optimal structure of the energy balance encourage the world to show a serious interest in finding an alternative to oil in order to stimulate its savings, as well as in making laws in energy saving. Finally, some considerations are expressed as to how cooperation can help the world avoid the occurrence of a catastrophic deficit during this transition period.

The concept of equilibrium is one of the most versatile in natural sciences. It applies to any system, whether it is a system of planets moving in stationary orbits around the star, or a population of tropical fish in the atoll lagoon. But the easiest way to understand the concept of the equilibrium state of the system on the example of mechanical systems. In the mechanics it is believed that the system is in equilibrium, if all the forces acting on it are completely balanced, that is, quench each other. If you read this book, for example, sitting in a chair, then you are in a state of equilibrium, since the power of the earthly attraction, pulling you down, is completely compensated for the power of the chair pressure on your body acting from the bottom up. You do not fall and do not take off precisely because you are in a state of equilibrium.

There are three types of equilibrium, corresponding to three physical situations.

Sustainable equilibrium

It is his most people usually understand "equilibrium". Imagine a bowl at the bottom of the spherical bowl. In a state of rest, it is strictly in the center of the bowl, where the action of the force of the gravitational attraction of the Earth is balanced by the power of the support reaction, directed strictly up, and the ball rests there as you are resting in your chair. If you shift the ball away from the center, throwing off his sideways and upwards in the direction of the bowl of the bowl, then it is worth release it, as he will immediately rush back to the deepest point in the center of the bowl - in the direction of the position of sustainable equilibrium.

You sitting in a chair, are at rest due to the fact that the system consisting of your body and armchair is in a state of stable equilibrium. Therefore, when changing some parameters of this system - for example, with an increase in your weight, if, suppose you, the child sat down on his knees, - a chair, being a material object, will change its configuration in such a way that the reaction force support will increase, - and you will remain In the position of a steady equilibrium (most of what can happen, the pillow will be burned down a little deeper).

In nature there are many examples of sustainable equilibrium in various systems (and not only mechanical). Consider, for example, the relationship of the predator sacrifice in the ecosystem. The ratio of the numbers of closed populations of predators and their victims quickly comes to an equilibrium state - so much hares in the forest from year to year consistently falls on so much fox, conventionally speaking. If for some reason, the population of the victims changes sharply (due to a burst of the fertility of the Zaitsev, for example), ecological equilibrium will be very soon restored due to the rapid growth of the population of predators, which will begin to exterminate the hares at the rate and They will not start dying away from hunger, leading to normal and its own livestock, resulting in the number of populations and hares, and the fox will come to the norm, which was observed before a burst of fertility at the hare. That is, in a sustainable ecosystem also acts internal forces (although not in the physical understanding of this word), seeking to return the system into a state of sustainable equilibrium in the event of a system deviation from it.

Similar effects can be observed in economic systems. A sharp drop in the price of goods leads to a surge of demand from hunters for cheap, subsequent reduction in inventories and, as a result, price increases and fall in demand for goods - and so until the system returns to the state of sustainable price equilibrium demand and supply. (Naturally, in real systems, and in environmental, and in economic, external factors, deflecting the system from an equilibrium state, for example, a seasonal fox shooting and / or hare or state-owned regulation and / or quotation of consumption. Such intervention leads to displacement Equilibrium, analogue of which in the mechanics will be, for example, deformation or tilt bowl.)

Unstable equilibrium

Not any equilibrium, however, is stable. Imagine a ball balancing on a knife blade. Directed strictly down the force of earthly attraction in this case is obviously also completely balanced by the directional support of the support reaction. But it is worth deflecting the center of the ball away from the point of resting on the blade line at least to a fraction of a millimeter (and for this there is a sufficiently meager power), as an equilibrium will be instantly violated and the power of earthly attraction will begin to carry the ball further from him.

An example of an unstable natural equilibrium is the thermal balance of the Earth when changing the global warming periods with new glacial periods and vice versa ( cm. Milankovich cycles). The average annual surface temperature of our planet is determined by the energy balance between the total solar radiation reaching the surface, and the total thermal radiation of the Earth into the outer space. Unstable this thermal balance becomes as follows. In some winter drops more snow than usual. The next summer, the heat is missing to melt the surplus of snow, and the summer is also colder than the usual due to the fact that due to the oversupply of the snow, the surface of the earth reflects back into space a large proportion of sunlight than before. Because of this, the next winter turns out to be even more snowy and cold than the previous one, and next summer on her surface there is even more snow and ice reflecting solar energy in space ... It is not difficult to see that the more such a global climate system deviates From the original point of thermal equilibrium, the faster the processes that lead the climate are growing further from it. Ultimately, on the surface of the Earth, multi-kilometer glaciers, which are inexorably moving in the direction of increasingly low latitudes, are formed on the surface of the Earth in the supremacy over the long years of global cooling. So it is difficult to imagine a more sharp balance than global-climatic.

Special mention deserves a type of unstable equilibrium, called metastableor quasi-resistant equilibrium. Imagine a bowl in a narrow and shallow groove - for example, on a rotated edge of the blade of a figure skate. Minor - per millimeter - the deviation from the point of equilibrium will lead to the emergence of the forces that the ball will return to the equilibrium state in the center of the groove. However, just more power is enough to bring the ball outside the zone of the metastable equilibrium, and it will fall from the blade of the skate. Metastable systems, as a rule, have a property to stay for a while in a state of equilibrium, after which they are "broken" from it as a result of any fluctuation of external influences and "fall" into an irreversible process characteristic of unstable systems.

A typical example of a quasi-resistant equilibrium is observed in the atoms of the working substance of some types of laser installations. Electrons in the atoms of the laser's working fluid occupy metastable atomic orbits and remain on them until the span of the first luminous quantum, which "knocks" them from the metastable orbit to a lower stable, emitting the new quantum of light coherent, which, in turn, The electro the next atom is knocked down with a metastable orbit, etc. The resulting avalanche reaction of radiation of coherent photons forming a laser beam, which, in fact, is based on the action of any laser.