Which equilibrium is called stable? III

The concept of equilibrium is one of the most universal in the natural sciences. It applies to any system, be it a system of planets moving in stationary orbits around a star, or a population of tropical fish in an atoll lagoon. But the easiest way to understand the concept of an equilibrium state of a system is through the example of mechanical systems. In mechanics, a system is considered to be in equilibrium if all the forces acting on it are completely balanced with each other, that is, they cancel each other out. If you are reading this book, for example, sitting in a chair, then you are in a state of equilibrium, since the force of gravity pulling you down is completely compensated by the force of pressure of the chair on your body, acting from the bottom up. You don't fall and you don't fly up precisely because you are in a state of balance.

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

Stable balance

This is what most people usually understand by “balance”.

Imagine a ball at the bottom of a spherical bowl. At rest, it is located strictly in the center of the bowl, where the action of the gravitational attraction of the Earth is balanced by the reaction force of the support, directed strictly upward, and the ball rests there just as you rest in your chair. If you move the ball away from the center, rolling it sideways and up towards the edge of the bowl, then as soon as you release it, it will immediately rush back to the deepest point in the center of the bowl - in the direction of the stable equilibrium position.

In nature there are many examples of stable equilibrium in various systems (and not only mechanical ones). Consider, for example, predator-prey relationships in an ecosystem. The ratio of the numbers of closed populations of predators and their victims quickly comes to an equilibrium state - so many hares in the forest from year to year stably account for so many foxes, relatively speaking. If for some reason the population size of the prey changes sharply (due to a surge in the birth rate of hares, for example), the ecological balance will very soon be restored due to the rapid increase in the number of predators, which will begin to exterminate the hares at an accelerated pace until the number of hares returns to normal and will not begin to die out from hunger themselves, bringing their own population back to normal, as a result of which the population numbers of both hares and foxes will return to the norm that was observed before the surge in the birth rate among hares. That is, in a stable ecosystem, internal forces also operate (although not in the physical sense of the word), seeking to return the system to a state of stable equilibrium if the system deviates from it.

Similar effects can be observed in economic systems. A sharp drop in the price of a product leads to a surge in demand from bargain hunters, a subsequent reduction in inventory and, as a consequence, an increase in price and a drop in demand for the product - and so on until the system returns to a state of stable price equilibrium of supply and demand. (Naturally, in real systems, both ecological and economic, external factors may act that deviate the system from an equilibrium state - for example, seasonal shooting of foxes and/or hares or government price regulation and/or consumption quotas. Such interference leads to a shift equilibrium, the analogue of which in mechanics would be, for example, the deformation or tilt of a bowl.)

Unstable equilibrium

Not every equilibrium, however, is stable. Imagine a ball balancing on a knife blade. The force of gravity directed strictly downward in this case is obviously also completely balanced by the force of the support reaction directed upward. But as soon as the center of the ball is deflected away from the rest point falling on the line of the blade even by a fraction of a millimeter (and for this a meager force influence is enough), the balance will be instantly disrupted and the force of gravity will begin to drag the ball further and further away from it.

An example of an unstable natural balance is the heat balance of the Earth when periods of global warming alternate with new ice ages and vice versa ( cm. Milankovitch 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 outer space. This heat balance becomes unstable in the following way. Some winters there is more snow than usual. The next summer there is not enough heat to melt the excess snow, and the summer is also colder than usual due to the fact that, due to the excess snow, the Earth's surface reflects a larger share of the sun's rays back into space than before. Because of this, the next winter turns out to be even snowier and colder than the previous one, and the following summer leaves even more snow and ice on the surface, reflecting solar energy into space... It is not difficult to see that the more such a global climate system deviates from the starting point of thermal equilibrium, the faster the processes that take the climate further away from it grow. Ultimately, on the surface of the Earth in the polar regions, over many years of global cooling, many kilometers of layers of glaciers are formed, which inexorably move towards lower and lower latitudes, bringing with them the next ice age to the planet. So it is difficult to imagine a more precarious balance than the global climate one.

A type of unstable equilibrium called metastable, or quasi-stable equilibrium. Imagine a ball in a narrow and shallow groove - for example, on the blade of a figure skate turned point up. A slight deviation - a millimeter or two - from the equilibrium point will lead to the emergence of forces that will return the ball to an equilibrium state in the center of the groove. However, a little more force will be enough to move the ball beyond the zone of metastable equilibrium, and it will fall off the blade of the skate. Metastable systems, as a rule, have the property of remaining in a state of equilibrium for some time, after which they “break away” from it as a result of any fluctuation in external influences and “collapse” into an irreversible process characteristic of unstable systems.

A typical example of quasi-stable equilibrium is observed in the atoms of the working substance of certain types of laser installations.

Electrons in the atoms of the laser working fluid occupy metastable atomic orbits and remain on them until the passage of the first light quantum, which “knocks” them from a metastable orbit to a lower stable one, emitting a new quantum of light, coherent to the passing one, which, in turn, knocks the electron of the next atom out of a metastable orbit, etc. As a result, an avalanche-like reaction of radiation of coherent photons is launched, forming a laser beam, which, in fact, underlies the action of any laser.

Equilibrium is a state of a system in which the forces acting on the system are balanced with each other. Equilibrium can be stable, unstable or indifferent.

The concept of equilibrium is one of the most universal in the natural sciences. It applies to any system, be it a system of planets moving in stationary orbits around a star, or a population of tropical fish in an atoll lagoon. But the easiest way to understand the concept of an equilibrium state of a system is through the example of mechanical systems. In mechanics, a system is considered to be in equilibrium if all the forces acting on it are completely balanced with each other, that is, they cancel each other out. If you are reading this book, for example, sitting in a chair, then you are in a state of equilibrium, since the force of gravity pulling you down is completely compensated by the force of pressure of the chair on your body, acting from the bottom up. You neither fall nor rise precisely because you are in a state of balance.

Stable balance

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

You, sitting in a chair, are in a state of rest due to the fact that the system consisting of your body and the chair is in a state of stable equilibrium. Therefore, when some parameters of this system change - for example, when your weight increases, if, say, a child sits on your lap - the chair, being a material object, will change its configuration in such a way that the force of the support reaction increases - and you will remain in a position of stable equilibrium (the most that can happen is that the pillow under you will sink a little deeper).

In nature there are many examples of stable equilibrium in various systems (and not only mechanical ones). Consider, for example, the predator-prey relationship in an ecosystem. The ratio of the numbers of closed populations of predators and their victims quickly comes to an equilibrium state - so many hares in the forest from year to year there are consistently so many foxes, relatively speaking. If for some reason the population size of the prey changes sharply (due to a surge in the birth rate of hares, for example), the ecological balance will very soon be restored due to the rapid increase in the number of predators, which will begin to exterminate the hares at an accelerated pace until the number of hares returns to normal and will not begin to die out from hunger themselves, bringing their own population back to normal, as a result of which the population numbers of both hares and foxes will return to the norm that was observed before the surge in the birth rate among hares. That is, in a stable ecosystem, internal forces also operate (although not in the physical sense of the word), seeking to return the system to a state of stable equilibrium if the system deviates from it.

Unstable equilibrium

A typical example of quasi-stable equilibrium is observed in the atoms of the working substance of certain types of laser installations. Electrons in the atoms of the laser working fluid occupy metastable atomic orbits and remain on them until the passage of the first light quantum, which “knocks” them from a metastable orbit to a lower stable one, emitting a new quantum of light, coherent to the passing one, which, in turn, knocks the electron of the next atom out of a metastable orbit, etc. As a result, an avalanche-like reaction of radiation of coherent photons is launched, forming a laser beam, which, in fact, underlies the action of any laser.

Indifferent Equilibrium

An intermediate case between stable and unstable equilibrium is the so-called indifferent equilibrium, in which any point in the system is an equilibrium point, and the deviation of the system from the initial point of rest does not change anything in the balance of forces within it. Imagine a ball on a completely smooth horizontal table - no matter where you move it, it will remain in a state of equilibrium.

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Lesson objectives: Study the state of balance of bodies, get acquainted with different types of balance; find out the conditions under which the body is in equilibrium.

Lesson objectives:

Educational: Study two conditions of equilibrium, types of equilibrium (stable, unstable, indifferent). Find out under what conditions bodies are more stable.
Educational: To promote the development of cognitive interest in physics. Development of skills to compare, generalize, highlight the main thing, draw conclusions.
Educational: To cultivate attention, the ability to express one’s point of view and defend it, to develop the communication abilities of students.

Lesson type: lesson on learning new material with computer support.

Equipment:

Disc “Work and Power” from “Electronic Lessons and Tests.
Table "Equilibrium conditions".
Tilting prism with plumb line.
Geometric bodies: cylinder, cube, cone, etc.
Computer, multimedia projector, interactive whiteboard or screen.
Presentation.

During the classes

Today in the lesson we will learn why the crane does not fall, why the Vanka-Vstanka toy always returns to its original state, why the Leaning Tower of Pisa does not fall?

I. Repetition and updating of knowledge.

State Newton's first law. What condition does the law refer to?
What question does Newton's second law answer? Formula and formulation.
What question does Newton's third law answer? Formula and formulation.
What is the resultant force? How is she located?
From the disk “Motion and interaction of bodies” complete task No. 9 “Resultant of forces with different directions” (the rule for adding vectors (2, 3 exercises)).

II. Learning new material.

1. What is called equilibrium?

Balance is a state of rest.

2. Equilibrium conditions.(slide 2)

a) When is the body at rest? What law does this follow from?

First equilibrium condition: A body is in equilibrium if the geometric sum of external forces applied to the body is equal to zero. ∑F = 0

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

Will it be in balance? (No, she will turn)

Only the central point is at rest, the rest are moving. This means that for a body to be in equilibrium, it is necessary that the sum of all forces acting on each element equals 0.

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

∑ M clockwise = ∑ M counterclockwise

Moment of force: M = F L

L – arm of force – the shortest distance from the fulcrum to the line of action of the force.

3. The center of gravity of the body and its location.(slide 4)

Body center of gravity- this is the point through which the resultant of all parallel forces of gravity acting on individual elements of the body passes (for any position of the body in space).

Find the center of gravity of the following figures:

4. Types of balance.

A) (slides 5–8)

Conclusion: Equilibrium is stable if, with a small deviation from the equilibrium position, there is a force tending to return it to this position.

The position in which its potential energy is minimal is stable. (slide 9)

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

Conclusion: For the stability of a body located at one point or line of support, it is necessary that the center of gravity be below the point (line) of support.

c) Stability of bodies located on a flat surface.

(slide 18)

1) Support surface– this is not always the surface that is in contact with the body (but the one that is limited by the lines connecting the legs of the table, tripod)

2) Analysis of the slide from “Electronic lessons and tests”, disk “Work and power”, lesson “Types of balance”.

Picture 1.

How are the stools different? (Support area)
Which one is more stable? (With larger area)
How are the stools different? (Location of the center of gravity)
Which one is the most stable? (Which center of gravity is lower)
Why? (Because it can be tilted to a larger angle without tipping over)

3) Experiment with a deflecting prism

Let's put a prism with a plumb line on the board and begin to gradually lift it by one edge. What do we see?
As long as the plumb line intersects the surface bounded by the support, equilibrium is maintained. But as soon as the vertical line passing through the center of gravity begins to go beyond the boundaries of the support surface, the whatnot tips over.

Analysis slides 19–22.

Conclusions:

The body that has the largest support area is stable.
Of two bodies of the same area, the one whose center of gravity is lower is stable, because it can be tilted without tipping over at a large angle.

Analysis slides 23–25.

Which ships are the most stable? Why? (In which the cargo is located in the holds, and not on the deck)

Which cars are the most stable? Why? (To increase the stability of cars when turning, the road surface is tilted in the direction of the turn.)

Conclusions: Equilibrium can be stable, unstable, indifferent. The greater the support area and the lower the center of gravity, the greater the stability of bodies.

III. Application of knowledge about the stability of bodies.

Which specialties are most in need of knowledge about body balance?
Designers and constructors of various structures (high-rise buildings, bridges, television towers, etc.)
Circus performers.
Drivers and other professionals.

(slides 28–30)

Why does “Vanka-Vstanka” return to the equilibrium position at any tilt of the toy?
Why does the Leaning Tower of Pisa stand at an angle and not fall?
How do cyclists and motorcyclists maintain balance?

Conclusions from the lesson:

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

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

Sources and literature used:

G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics. Grade 10.
Filmstrip “Sustainability” 1976 (scanned by me on a film scanner).
Disc “Motion and interaction of bodies” from “Electronic lessons and tests”.
Disc "Work and Power" from "Electronic Lessons and Tests".

Mechanical balance

Mechanical balance- a state of a mechanical system in which the sum of all forces acting on each of its particles is equal to zero and the sum of the moments of all forces applied to the body relative to any arbitrary axis of rotation is also zero.

In a state of equilibrium, the body is at rest (the velocity vector is zero) in the chosen reference frame, either moves uniformly in a straight line or rotates without tangential acceleration.

Definition through system energy

Since energy and forces are related by fundamental relationships, this definition is equivalent to the first. However, the definition in terms of energy can be extended to provide information about the stability of the equilibrium position.

Types of balance

Let's give an example for a system with one degree of freedom. In this case, a sufficient condition for the equilibrium position will be the presence of a local extremum at the point under study. As is known, the condition for a local extremum of a differentiable function is that its first derivative is equal to zero. To determine when this point is a minimum or maximum, you need to analyze its second derivative. The stability of the equilibrium position is characterized by the following options:

unstable equilibrium;
stable balance;
indifferent equilibrium.

Unstable equilibrium

In the case when the second derivative is negative, the potential energy of the system is in a state of local maximum. This means that the equilibrium position unstable. If the system is displaced a small distance, it will continue its movement due to the forces acting on the system.

Stable balance

Second derivative > 0: potential energy at local minimum, equilibrium position sustainable(see Lagrange's theorem on the stability of equilibrium). If the system is displaced a small distance, it will return back to its equilibrium state. Equilibrium is stable if the center of gravity of the body occupies the lowest position compared to all possible neighboring positions.

Indifferent Equilibrium

Second derivative = 0: in this region the energy does not vary and the equilibrium position is indifferent. If the system is moved a small distance, it will remain in the new position.

Stability in systems with a large number of degrees of freedom

If a system has several degrees of freedom, then it may turn out that in shifts in some directions the equilibrium is stable, but in others it is unstable. The simplest example of such a situation is a “saddle” or “pass” (it would be good to place a picture in this place).

The equilibrium of a system with several degrees of freedom will be stable only if it is stable in all directions.

Wikimedia Foundation.

2010.

See what “Mechanical balance” is in other dictionaries: mechanical balance

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A market equilibrium is called stable if, when it deviates from the equilibrium state, market forces come into play and restore it. Otherwise, the equilibrium is unstable.

To check whether the situation presented in Fig. 4.7, stable equilibrium, let us assume that the price increased from R 0 to P 1. As a result, a surplus in the amount of Q2 – Q1 is formed on the market. There are two versions about what will happen after this: L. Walras and A. Marshall.

According to L. Walras, when there is an excess, competition arises between sellers. To attract buyers, they will begin to reduce the price. As the price decreases, the quantity demanded will increase and the quantity supplied will decrease until the original equilibrium is restored. If the price deviates downward from its equilibrium value, demand will exceed supply. Competition will begin between buyers

Rice. 4.7. Restoring balance. Pressure: 1 – according to Marshall; 2 – according to Walras

for scarce goods. They will offer sellers a higher price, which will increase supply. This will continue until the price returns to the equilibrium level P0. Therefore, according to Walras, the combination P0, Q0 represents a stable market equilibrium.

A. Marshall reasoned differently. When the quantity supplied is less than the equilibrium value, then the demand price exceeds the supply price. Firms make a profit, which stimulates the expansion of production, and the quantity supplied will increase until it reaches the equilibrium value. If the supply exceeds the equilibrium volume, the demand price will be lower than the supply price. In such a situation, entrepreneurs incur losses, which will lead to a reduction in production to the equilibrium break-even volume. Consequently, according to Marshall, the point of intersection of the supply and demand curves in Fig. 4.7 represents a stable market equilibrium.

According to L. Walras, in conditions of shortage the active side of the market is buyers, and in conditions of excess – sellers. According to A. Marshall, entrepreneurs are always the dominant force in shaping market conditions.

However, the two considered options for diagnosing the stability of market equilibrium lead to the same result only in cases of a positive slope of the supply curve and a negative slope of the demand curve. When this is not the case, then the diagnosis of the stability of equilibrium market states according to Walras and Marshall do not coincide. Four variants of such states are shown in Fig. 4.8.

Rice. 4.8.

The situations presented in Fig. 4.8, a, V, possible under conditions of increasing economies of scale, when producers can reduce the supply price as output increases. The positive slope of the demand curve in the situations shown in Fig. 4.8, b, d, may reflect the Giffen paradox or the snob effect.

According to Walras, the sectoral equilibrium presented in Fig. 4.8, a, b, is unstable. If the price rises to R 1, then there will be a shortage in the market: QD > QS. In such conditions, buyer competition will cause further price increases. If the price drops to P0, then supply will exceed demand, which, according to Walras, should lead to a further decrease in price. According to Marshall combination P*, Q* represents a stable equilibrium. If the supply is less than Q*, the demand price will be higher than the supply price, and this stimulates an increase in output. If Q* increases, the demand price will be lower than the supply price, so it will decrease.

When the supply and demand curves are located as shown in Fig. 4.8, c, d, then, according to Walrasian logic, equilibrium is at the point P*, Q* is stable, since at P1 > P* an excess occurs, and at 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* is the opposite.

The discrepancies between L. Walras and A. Marshall in describing the mechanism of market functioning are caused by the fact that, according to the first, market prices are completely flexible and instantly respond to any changes in the market situation, and according to the second, prices are not flexible enough even when imbalances arise between demand and supply, the volumes of market transactions respond to them faster than prices. The interpretation of the process of establishing market equilibrium according to Walras corresponds to the conditions of perfect competition, and according to Marshall - to imperfect competition in a short period.

L. Walras (1834–1910) – founder of the concept of general economic equilibrium.