The branch of mechanics in which the conditions for the equilibrium of bodies are studied is called statics. It follows from Newton's second law that if the vector sum of all forces applied to a body is zero, then the body keeps its speed unchanged. In particular, if the initial velocity is zero, the body remains at rest. The condition of the invariance of the speed of the body can be written as:

or in projections on the coordinate axes:

.

It is obvious that a body can only be at rest with respect to one particular coordinate system. In statics, the equilibrium conditions of bodies are studied precisely in such a system. The necessary equilibrium condition can also be obtained by considering the motion of the center of mass of a system of material points. Internal forces do not affect the movement of the center of mass. The acceleration of the center of mass is determined by the vector sum of the external forces. But if this sum is equal to zero, then the acceleration of the center of mass, and, consequently, the speed of the center of mass. If at the initial moment , then the center of mass of the body remains at rest.

Thus, the first condition for the equilibrium of bodies is formulated as follows: the speed of the body does not change if the sum of external forces applied at each point is equal to zero. The resulting rest condition for the center of mass is a necessary (but not sufficient) condition for the equilibrium of a rigid body.

Example

It may be that all the forces acting on the body are balanced, however, the body will accelerate. For example, if you apply two equal and oppositely directed forces (they are called a pair of forces) to the center of mass of the wheel, then the wheel will be at rest if its initial speed was zero. If these forces are applied to different points, then the wheel will begin to rotate (Fig. 4.5). This is because the body is in equilibrium when the sum of all forces is zero at every point of the body. But if the sum of external forces is equal to zero, and the sum of all forces applied to each element of the body is not equal to zero, then the body will not be in equilibrium, possibly (as in the example considered) rotational motion. Thus, if a body can rotate about a certain axis, then for its equilibrium it is not enough that the resultant of all forces be equal to zero.

To obtain the second equilibrium condition, we use the equation of rotational motion , where is the sum of the moments of external forces about the axis of rotation. When , then b = 0, which means that the angular velocity of the body does not change . If at the initial moment w = 0, then the body will not rotate further. Consequently, the second condition for mechanical equilibrium is the requirement that the algebraic sum of the moments of all external forces about the axis of rotation be equal to zero:

In the general case of an arbitrary number of external forces, the equilibrium conditions can be represented as follows:

,

.

These conditions are necessary and sufficient.

Example

Equilibrium is stable, unstable and indifferent. The equilibrium is stable if, with small displacements of the body from the equilibrium position, the forces acting on it and the moments of forces tend to return the body to the equilibrium position (Fig. 4.6a). The equilibrium is unstable if the acting forces at the same time take the body even further from the equilibrium position (Fig. 4.6b). If, at small displacements of the body, the acting forces are still balanced, then the equilibrium is indifferent (Fig. 4.6c). A ball lying on a flat horizontal surface is in a state of indifferent equilibrium. A ball located at the top of a spherical ledge is an example of an unstable equilibrium. Finally, the ball at the bottom of the spherical cavity is in a state of stable equilibrium.

An interesting example of the equilibrium of a body on a support is the leaning tower in the Italian city of Pisa, which, according to legend, was used by Galileo when studying the laws of free fall of bodies. The tower has the shape of a cylinder with a radius of 7 m. The top of the tower is deviated from the vertical by 4.5 m.

The Leaning Tower of Pisa is famous for its steep slope. The tower is falling. The height of the tower is 55.86 meters from the ground on the lowest side and 56.70 meters on the highest side. Its weight is estimated at 14,700 tons. The current slope is about 5.5°. A vertical line drawn through the center of mass of the tower intersects the base approximately 2.3 m from its center. Thus, the tower is in a state of equilibrium. The balance will be disturbed and the tower will fall when the deviation of its top from the vertical reaches 14 m. Apparently, this will not happen very soon.

It was believed that the curvature of the tower was originally conceived by the architects - in order to demonstrate their outstanding skills. But something else is much more likely: the architects knew that they were building on an extremely unreliable foundation, and therefore laid in the design the possibility of a slight deviation.

When there was a real threat of the collapse of the tower, modern engineers took it up. It was pulled into a steel corset of 18 cables, the foundation was weighted with lead blocks and at the same time the soil was strengthened by pumping concrete underground. With the help of all these measures, it was possible to reduce the angle of inclination of the falling tower by half a degree. Experts say that now it will be able to stand for at least another 300 years. From the point of view of physics, the measures taken mean that the equilibrium conditions of the tower have become more reliable.

For a body with a fixed axis of rotation, all three types of equilibrium are possible. Indifferent equilibrium occurs when the axis of rotation passes through the center of mass. In stable and unstable equilibrium, the center of mass is on a vertical line passing through the axis of rotation. In this case, if the center of mass is below the axis of rotation, the state of equilibrium is stable (Fig. 4.7a). If the center of mass is located above the axis, the equilibrium state is unstable (Fig. 4.7b).

A special case of equilibrium is the equilibrium of a body on a support. In this case, the elastic force of the support is not applied to one point, but is distributed over the base of the body. The body is in equilibrium if a vertical line drawn through the center of mass of the body passes through the support area, that is, inside the contour formed by lines connecting the support points. If this line does not cross the area of ​​support, then the body overturns.

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

Definition through the energy of the system[ | ]

Since energy and forces are connected by fundamental dependencies, this definition is equivalent to the first one. However, the definition in terms of energy can be extended in order to obtain information about the stability of the equilibrium position.

Types of balance [ | ]

There are three types of balance of bodies: stable, unstable and indifferent. The equilibrium is called stable if, after small external influences, the body returns to its original state of equilibrium. Equilibrium is called unstable if, with a slight displacement of the body from the equilibrium position, the resultant of the forces applied to it is nonzero and is directed from the equilibrium position. Equilibrium is called indifferent if, with a small displacement of the body from the equilibrium position, the resultant of the forces applied to it is equal to zero.

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 of the potential energy at the point under study. As is known, the condition for a local extremum of a differentiable function is the equality to zero of its first derivative . To determine when this point is a minimum or maximum, it is necessary to analyze its second derivative. The stability of the equilibrium position is characterized by the following options:

• unstable equilibrium;
• stable balance;
• indifferent balance.

Unstable equilibrium[ | ]

In the case when the second derivative is negative, the potential energy of the system is in the state of a local maximum. This means that the equilibrium position unstable. If the system is displaced by a small distance, then it will continue its movement due to the forces acting on the system. That is, when the body is taken out of balance, it does not return to its original position.

sustainable balance[ | ]

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

Indifferent balance[ | ]

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. If you deflect or move the body, it will remain in balance.

Stability in systems with a large number of degrees of freedom[ | ]

If the system has several degrees of freedom, then it may turn out that with deviations along a particular direction, the equilibrium is stable, but if the equilibrium is unstable in at least one direction, then it is also unstable in general. The simplest example of such a situation is an equilibrium point of the “saddle” or “pass” type.

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

All forces applied to the body about any arbitrary axis of rotation is also equal to zero.

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

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Subtitles

Definition through the energy of the system

Since energy and forces are connected by fundamental dependencies , this definition is equivalent to the first one. However, the definition in terms of energy can be extended in order to obtain information about the stability of the equilibrium position.

Types of balance

Let us 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 the equality to zero of its first derivative . To determine when this point is a minimum or maximum, it is necessary to analyze its second derivative. The stability of the equilibrium position is characterized by the following options:

• unstable equilibrium;
• stable balance;
• indifferent balance.

In the case when the second derivative is negative, the potential energy of the system is in the state of a local maximum. This means that the equilibrium position unstable. If the system is displaced by a small distance, then it will continue its movement due to the forces acting on the system. That is, when the body is taken out of balance, it does not return to its original position.

sustainable balance

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

Indifferent balance

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. If you deflect or move the body, it will remain in balance.

• Types of sustainability

Equilibrium is a state of the 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, whether it be 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 the equilibrium state of a system is by the example of mechanical systems. In mechanics, it is considered that the system is 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, while sitting in a chair, then you are just in a state of balance, since the force of gravity pulling you down is completely compensated by the pressure of the chair on your body, acting from the bottom up. You don’t fall down and take off precisely because you are in a state of balance.

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

sustainable 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 force of the gravitational attraction of the Earth is balanced by the reaction force of the support directed strictly upwards, and the ball rests there just like you rest in your chair. If you move the ball away from the center, rolling it sideways and upwards towards the edge of the bowl, then, as soon as you release it, it immediately rushes back to the deepest point in the center of the bowl - in the direction of the position of stable equilibrium.

You, sitting in a chair, are at rest due to the fact that the system consisting of your body and chair is in a state of stable equilibrium. Therefore, when some parameters of this system change - for example, when you increase your weight, if, let's say, a child sits on your lap - the chair, being a material object, will change its configuration in such a way that the reaction force of the support will increase - and you will remain in a position of stable balance (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 prey quickly comes to an equilibrium state - so many hares in the forest from year to year steadily account for so many foxes, relatively speaking. If for some reason the population of prey changes dramatically (due to a surge in the birth rate of hares, for example), the ecological balance will be restored very soon due to the rapid increase in the number of predators, which will begin to exterminate hares at an accelerated pace until they bring the number of hares back to normal and they will not begin to die of hunger themselves, bringing their own livestock back to normal, as a result of which the populations of both hares and foxes will return to the norm that was observed before the surge in the birth rate of 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 in case the system deviates from it.

Similar effects can be observed in economic systems. A sharp drop in the price of a good leads to a surge in demand from bargain hunters, a subsequent reduction in inventories and, as a result, an increase in price and a drop in demand for the good - 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, there may be external factors that deviate the system from the equilibrium state - for example, seasonal shooting of foxes and / or hares or state price regulation and / or consumption quotas. Such intervention leads to a bias equilibrium, whose analogue in mechanics would be, for example, the deformation or inclination of the bowl.)

Unstable equilibrium

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

An example of an unstable natural equilibrium is the heat balance of the Earth when periods of global warming are replaced by 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 as follows. Some winters get more snow than usual. The following 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 of snow, the surface of the Earth reflects back into space a greater proportion of the sun's rays than before. Because of this, the next winter turns out to be even snowier and colder than the previous one, and the following summer, even more snow and ice remain on the surface, reflecting solar energy into space ... It is easy 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 even further away from it increase. Ultimately, on the surface of the Earth in the polar regions, for many years of global cooling, many kilometers of strata of glaciers are formed, which inexorably move towards ever lower latitudes, bringing with them another ice age to the planet. So it is difficult to imagine a more precarious balance than global climate.

Of particular note is a kind 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 upside down. A slight - by a millimeter or two - deviation 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 is enough to take the ball out of the zone of metastable equilibrium, and it will fall off the skate blade. Metastable systems, as a rule, have the property of staying in a state of equilibrium for some time, after which they "break" out of it as a result of some fluctuation of external influences and "fall" into an irreversible process characteristic of unstable systems.

A typical example of a quasi-stable equilibrium is observed in the atoms of the working substance of some types of laser systems. The electrons in the atoms of the working body of the laser occupy metastable atomic orbits and remain on them until the passage of the first light quantum, which "knocks" them from the metastable orbit to a lower stable one, while emitting a new light quantum, coherent to the passing one, which, in turn, knocks down the electron of the next atom from the metastable orbit, etc. As a result, an avalanche-like reaction of emission of coherent photons forming a laser beam is launched, which, in fact, underlies the operation of any laser.

Indifferent balance

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

The branch of mechanics in which the conditions for the equilibrium of bodies are studied is called statics. The easiest way is to consider the equilibrium conditions for an absolutely rigid body, i.e., such a body, the dimensions and shape of which can be considered unchanged. The concept of an absolutely rigid body is an abstraction, since all real bodies, under the influence of forces applied to them, are deformed to one degree or another, that is, they change their shape and size. The magnitude of deformations depends both on the forces applied to the body and on the properties of the body itself - its shape and the properties of the material from which it is made. In many practically important cases, the deformations are small, and the use of the concepts of an absolutely rigid body is justified.

Model of a perfectly rigid body. However, the smallness of deformations is not always a sufficient condition for a body to be considered absolutely rigid. To clarify this, consider the following example. A board resting on two supports (Fig. 140a) can be considered as an absolutely rigid body, despite the fact that it slightly bends under the influence of gravity. Indeed, in this case, the conditions of mechanical equilibrium make it possible to determine the reaction forces of the supports without taking into account the deformation of the board.

But if the same board lies on the same supports (Fig. 1406), then the idea of ​​an absolutely rigid body is inapplicable. Indeed, let the extreme supports be on the same horizontal line, and the middle one a little lower. If the board is absolutely solid, that is, it does not bend at all, then it does not put pressure on the middle support at all. If the board bends, then it presses on the middle support, and the stronger, the greater the deformation. Terms

The equilibrium of an absolutely rigid body in this case does not allow determining the reaction forces of the supports, since they lead to two equations for three unknown quantities.

Rice. 140. Reaction forces acting on a board lying on two (a) and three (b) supports

Such systems are called statically indeterminate. To calculate them, it is necessary to take into account the elastic properties of bodies.

The given example shows that the applicability of the absolutely rigid body model in statics is determined not so much by the properties of the body itself, but by the conditions in which it is located. So, in the example considered, even a thin straw can be considered an absolutely solid body if it lies on two supports. But even a very rigid beam cannot be considered an absolutely rigid body if it rests on three supports.

Equilibrium conditions. The equilibrium conditions for an absolutely rigid body are a special case of dynamic equations when there is no acceleration, although historically statics arose from the needs of construction equipment almost two millennia earlier than dynamics. In an inertial frame of reference, a rigid body is in equilibrium if the vector sum of all external forces acting on the body and the vector sum of the moments of these forces are equal to zero. When the first condition is met, the acceleration of the center of mass of the body is equal to zero. When the second condition is met, there is no angular acceleration of rotation. Therefore, if at the initial moment the body was at rest, then it will remain at rest further.

In what follows, we confine ourselves to the study of relatively simple systems in which all acting forces lie in the same plane. In this case, the vector condition

reduces to two scalars:

if the axes of the plane of action of forces are located. Some of the external forces entering into the equilibrium conditions (1) acting on the body can be given, i.e., their modules and directions are known. As for the reaction forces of bonds or supports that limit the possible movement of the body, they, as a rule, are not predetermined and are themselves subject to determination. In the absence of friction, the reaction forces are perpendicular to the contact surface of the bodies.

Rice. 141. To determine the direction of reaction forces

reaction forces. Sometimes doubts arise in determining the direction of the bond reaction force, as, for example, in Fig. 141, which shows a rod resting at point A on the smooth concave surface of the cup and at point B on the sharp edge of the cup.

To determine the direction of the reaction forces in this case, you can mentally move the rod a little without disturbing its contact with the cup. The reaction force will be directed perpendicular to the surface on which the contact point slides. So, at point A, the reaction force acting on the rod is perpendicular to the surface of the cup, and at point B, it is perpendicular to the rod.

Moment of power. Moment M of force relative to some point

O is called the vector product of the radius-vector drawn from O to the point of application of the force, by the force vector

The vector M of the moment of force is perpendicular to the plane in which the vectors lie

Equation of moments. If several forces act on the body, then the second equilibrium condition associated with the moments of forces is written as

In this case, the point O, from which the radius vectors are drawn, must be chosen common for all acting forces.

For a flat system of forces, the vectors of the moments of all forces are directed perpendicular to the plane in which the forces lie, if the moments are considered relative to a point lying in the same plane. Therefore, the vector condition (4) for the moments reduces to one scalar one: in the equilibrium position, the algebraic sum of the moments of all external acting forces is equal to zero. The module of the moment of force relative to the point O is equal to the product of the module

forces at a distance from point O to the line along which the force acts. In this case, the moments tending to rotate the body clockwise are taken with one sign, counterclockwise - with the opposite. The choice of the point relative to which the moments of forces are considered is made solely for reasons of convenience: the equation of moments will be the simpler, the more forces will have moments equal to zero.

Balance example. To illustrate the application of the equilibrium conditions for a perfectly rigid body, consider the following example. A light step-ladder consists of two identical parts, hinged at the top and tied with a rope at the base (Fig. 142). Let us determine what is the tension force of the rope, with what forces the halves of the ladder interact in the hinge, and with what forces they press on the floor, if a person of weight P is standing in the middle of one of them.

The system under consideration consists of two rigid bodies - ladder halves, and the equilibrium conditions can be applied both to the system as a whole and to its parts. By applying the equilibrium conditions to the entire system as a whole, one can find the reaction forces of the floor and (Fig. 142). In the absence of friction, these forces are directed vertically upwards, and the condition that the vector sum of external forces (1) is equal to zero takes the form

The equilibrium condition for the moments of external forces relative to point A is written as follows:

where - the length of the stairs, the angle formed by the stairs with the floor. Solving the system of equations (5) and (6), we find

Rice. 142. The vector sum of external forces and the sum of the moments of external forces in equilibrium is zero

Of course, instead of the equation of moments (6) with respect to point A, one could write the equation of moments with respect to point B (or any other point). This would result in a system of equations equivalent to the system (5) and (6) used.

The tension force of the rope and the interaction force in the hinge for the considered physical system are internal and therefore cannot be determined from the equilibrium conditions of the entire system as a whole. To determine these forces, it is necessary to consider the conditions for the equilibrium of individual parts of the system. Wherein

By a good choice of the point relative to which the equation of the moments of forces is compiled, it is possible to achieve a simplification of the algebraic system of equations. So, for example, in this system, we can consider the equilibrium condition for the moments of forces acting on the left half of the ladder, relative to point C, where the hinge is located.

With this choice of point C, the forces acting in the hinge will not enter into this condition, and we immediately find the tension force of the rope T:

whence, given that we get

Condition (7) means that the resultant of the forces T and passes through the point C, i.e., is directed along the stairs. Therefore, the equilibrium of this half of the ladder is possible only if the force acting on it in the hinge is also directed along the ladder (Fig. 143), and its modulus is equal to the modulus of the resultant forces T and

Rice. 143. The lines of action of all three forces acting on the left half of the stairs pass through one point

The absolute value of the force acting in the hinge on the other half of the stairs, based on Newton's third law, is equal to and its direction is opposite to the direction of the vector. The direction of the force could be determined directly from fig. 143, given that when a body is in equilibrium under the action of three forces, the lines along which these forces act intersect at one point. Indeed, consider the point of intersection of the lines of action of two of these three forces and draw up an equation of moments about this point. The moments of the first two forces about this point are equal to zero; hence, the moment of the third force must also be equal to zero, which, in accordance with (3), is possible only if the line of its action also passes through this point.

The golden rule of mechanics. Sometimes the problem of statics can be solved without considering the equilibrium conditions at all, but using the law of conservation of energy in relation to mechanisms without friction: no mechanism gives a gain in work. This law

called the golden rule of mechanics. To illustrate this approach, consider the following example: a heavy load of weight P is suspended on a weightless hinge with three links (Fig. 144). What tension must be maintained by the thread connecting points A and B?

Rice. 144. To the determination of the tension force of the thread in a three-link hinge supporting a load of weight P

Let's try using this mechanism to lift the load P. Having untied the thread at point A, we pull it up so that point B slowly rises a distance. This distance is limited by the fact that the tension force of the thread T must remain unchanged during the movement. In this case, as will be seen from the answer, the force T does not depend at all on how much the hinge is compressed or stretched. A job well done. As a result, the load P rises to a height which, as is clear from geometric considerations, is equal to. Since in the absence of friction no energy losses occur, it can be argued that the change in the potential energy of the load equal to is determined by the work done during lifting. That's why

Obviously, for a hinge containing an arbitrary number of identical links,

It is not difficult to find the tension force of the thread, and in the case when it is required to take into account the weight of the hinge itself, the work done during lifting should be equated to the sum of changes in the potential energies of the load and the hinge. For a hinge of identical links, its center of mass rises to Therefore

The formulated principle (“golden rule of mechanics”) is also applicable when there is no change in potential energy in the process of displacement, and the mechanism is used to transform the force. Gearboxes, transmissions, gates, systems of levers and blocks - in all such systems, the transformed force can be determined by equating the work of the transformed and applied forces. In other words, in the absence of friction, the ratio of these forces is determined only by the geometry of the device.

Consider from this point of view the above example with a stepladder. Of course, it is hardly advisable to use a stepladder as a lifting mechanism, i.e., to lift a person by bringing the halves of the stepladder together. However, this cannot prevent us from applying the method described to find the tension in the rope. Equating the work done when the parts of the stepladder come closer to the change in the potential energy of a person on the stepladder and connecting from geometric considerations the movement of the lower end of the ladder with the change in the height of the load (Fig. 145), we obtain, as expected, the result given earlier:

As already noted, the displacement should be chosen in such a way that the acting force can be considered constant during its process. It is easy to see that in the example with a hinge, this condition does not impose restrictions on movement, since the thread tension does not depend on the angle (Fig. 144). On the other hand, in the stepladder problem, the displacement should be chosen small, because the tension on the rope depends on the angle a.

Balance stability. Equilibrium is stable, unstable and indifferent. The equilibrium is stable (Fig. 146a), if, with small displacements of the body from the equilibrium position, the acting forces tend to return it back, and unstable (Fig. 1466), if the forces take it further from the equilibrium position.

Rice. 145. Movement of the lower ends of the ladder and the movement of cargo when the halves of the ladder approach each other

Rice. 146. Stable (a), unstable (b) and indifferent (c) equilibrium

If, at small displacements, the forces acting on the body and their moments are still balanced, then the equilibrium is indifferent (Fig. 146c). With indifferent equilibrium, the neighboring positions of the body are also in equilibrium.

Let us consider examples of the study of equilibrium stability.

1. A stable equilibrium corresponds to a minimum potential energy of the body in relation to its values ​​in neighboring positions of the body. It is often convenient to use this property in finding the equilibrium position and in studying the nature of the equilibrium.

Rice. 147. Stability of body balance and position of the center of mass

A vertical free-standing column is in stable equilibrium, since its center of mass rises at small inclinations. This happens until the vertical projection of the center of mass goes beyond the support area, i.e. the angle of deviation from the vertical does not exceed a certain maximum value. In other words, the region of stability extends from the minimum potential energy (in a vertical position) to the maximum closest to it (Fig. 147). When the center of mass is located exactly above the boundary of the area of ​​support, the column is also in equilibrium, but unstable. A horizontally lying column corresponds to a much wider region of stability.

2. There are two round pencils with radii and One of them is located horizontally, the other is balanced on it in a horizontal position so that the axes of the pencils are mutually perpendicular (Fig. 148a). At what ratio between the radii is the equilibrium stable? At what maximum angle can the top pencil be deflected from the horizontal? The coefficient of friction of pencils against each other is equal to

At first glance, it may seem that the balance of the upper pencil is generally unstable, since the center of mass of the upper pencil lies above the axis around which it can rotate. However, here the position of the rotation axis does not remain unchanged; therefore, this case requires a special study. Since the top pencil is balanced in a horizontal position, the centers of mass of the pencils lie on this vertical (Fig. ).

Deviate the top pencil at some angle from the horizontal. In the absence of static friction, it would immediately slide down. In order not to think about possible slippage for the time being, we will assume that the friction is sufficiently large. In this case, the upper pencil "rolls" along the lower one without slipping. The fulcrum from position A moves to a new position C, and the point that the upper pencil rested on the lower one before the deviation

moves to position B. Since there is no slip, the length of the arc is equal to the length of the segment

Rice. 148. The upper pencil is balanced in a horizontal position on the lower pencil (a); to the study of equilibrium stability (b)

The center of mass of the upper pencil moves to position . If the vertical drawn through passes to the left of the new fulcrum C, then gravity tends to return the upper pencil to its equilibrium position.

Let's express this condition mathematically. Drawing a vertical line through point B, we see that the condition must be satisfied

Since then from condition (8) we obtain

Since gravity will tend to return the upper pencil to the equilibrium position only at Therefore, stable equilibrium of the upper pencil on the lower one is possible only when its radius is less than the radius of the lower pencil.

The role of friction. To answer the second question, it is necessary to find out what reasons limit the permissible deviation angle. First, at large deflection angles, the vertical drawn through the center of mass of the upper pencil can pass to the right of the support point C. From condition (9) it can be seen that for a given ratio of pencil radii, the maximum deflection angle

Are the equilibrium conditions of a rigid body always sufficient to determine the reaction forces?

How can one practically determine the direction of the reaction forces in the absence of friction?

How can the golden rule of mechanics be used in the analysis of equilibrium conditions?

If in the hinge shown in Fig. 144, with a thread to connect not points A and B, but points L and C, then what will be its tension force?

How is the stability of the equilibrium of a system related to its potential energy?

What conditions determine the maximum angle of deflection of a body resting on a plane at three points so that its stability is not lost?