Equilibrium mechanical system- this is a state in which all points of the mechanical system are at rest in relation to the considered frame of reference. If the frame of reference is inertial, equilibrium is called absolute if non-inertial - relative.

To find the conditions for the equilibrium of an absolutely rigid body, it is necessary to mentally break it down into a large number of sufficiently small elements, each of which can be represented by a material point. All these elements interact with each other - these forces of interaction are called internal... In addition, external forces can act on a number of points on the body.

According to Newton's second law, for the acceleration of a point to be zero (and the acceleration of a stationary point to be zero), the geometric sum of the forces acting on this point must be equal to zero. If the body is at rest, then all its points (elements) are also at rest. Therefore, for any point of the body, you can write:

where is the geometric sum of all external and internal forces acting on i th element of the body.

The equation means that for the balance of a body it is necessary and sufficient that the geometric sum of all forces acting on any element of this body is equal to zero.

From it is easy to obtain the first condition for the balance of a body (system of bodies). To do this, it is enough to sum up the equation over all elements of the body:

.

The second sum is equal to zero according to Newton's third law: the vector sum of all internal forces of the system is equal to zero, since any internal force corresponds to a force equal in magnitude and opposite in direction.

Hence,

.

The first condition for equilibrium of a rigid body(body systems) is the equality to zero of the geometric sum of all external forces applied to the body.

This condition is necessary, but not sufficient. It is easy to verify this by remembering the rotating action of a pair of forces, the geometric sum of which is also equal to zero.

The second condition for the equilibrium of a rigid body is the equality to zero of the sum of the moments of all external forces acting on the body, relative to any axis.

Thus, the equilibrium conditions for a rigid body in the case of an arbitrary number of external forces are as follows:

.

Class: 10

Lesson presentation

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.

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 the bodies are more stable.
• Developing: Promote the development of a cognitive interest in physics. Development of skills to compare, generalize, highlight the main thing, draw conclusions.
• Educational: To educate attention, the ability to express their point of view and defend it, to develop the communication skills of students.

Lesson type: a lesson in learning new material with computer support.

Equipment:

1. Disk "Work and Power" from "Electronic Lessons and Tests.
2. Equilibrium conditions table.
3. Tilting prism with a plumb line.
4. Geometric bodies: cylinder, cube, cone, etc.
5. Computer, multimedia projector, interactive whiteboard or screen.
6. Presentation.

During the classes

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

I. Repetition and actualization of knowledge.

1. Formulate Newton's first law. What state does the law say?
2. What question does Newton's second law answer? Formula and wording.
3. What question does Newton's third law answer? Formula and wording.
4. What is called the resultant force? How is it located?
5. From the disc "Movement and interaction of bodies" complete task number 9 "Resultant forces with different directions" (the rule of adding vectors (2, 3 exercises)).

II. Learning new material.

1. What is called balance?

Equilibrium is a state of calm.

2. Equilibrium conditions.(slide 2)

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

The first equilibrium condition: The body is in equilibrium if the geometric sum of the 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 she be in balance? (No, she will turn)

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

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

∑ M clockwise = ∑ M counterclockwise

Moment of force: M = F L

L - shoulder 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 gravity forces acting on individual elements of the body passes (for any position of the body in space).

Find the center of gravity of the following shapes:

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.

Stable is the position in which its potential energy is minimal. (slide 9)

b) Stability of bodies located on the fulcrum or on the line of support.(slides 10-17)

Conclusion: For the stability of a body located on 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 on a flat surface.

(slide 18)

1) Support surface- it is not always the surface that is in contact with the body (but the one that is bounded by the lines connecting the legs of the table, tripods)

2) Analysis of the slide from "Electronic lessons and tests", disc "Work and power", lesson "Types of balance".

Picture 1.

1. How are stools different? (Support area)
2. Which one is more stable? (With a larger area)
3. How are stools different? (The location of the center of gravity)
4. Which one is the most stable? (With lower center of gravity)
5. Why? (Since it can be tilted to a larger angle without overturning)

3) Experiment with a deflecting prism

1. We put a prism with a plumb line on the board and begin to gradually raise it over one edge. What do we see?
2. As long as the plumb line crosses the surface bounded by the support, balance is maintained. But as soon as the vertical, passing through the center of gravity, begins to go beyond the boundaries of the support surface, the stack overturns.

Parsing slides 19-22.

Conclusions:

1. The body with a larger support area is stable.
2. Of two bodies of the same area, the one with the lower center of gravity is stable. it can be tilted without overturning at a large angle.

Parsing slides 23-25.

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

Which cars are the most resilient? Why? (To increase the stability of cars on bends, the roadbed is tilted towards the bend.)

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

III. Application of knowledge about the stability of bodies.

1. What specialties are most needed to know about the balance of bodies?
2. Designers and constructors of various structures (high-rise buildings, bridges, television towers, etc.)
3. Circus artists.
4. Drivers and other professionals.

(slides 28-30)

1. Why does Vanka-Vstanka return to the balance position at any tilt of the toy?
2. Why is the Leaning Tower of Pisa tilted and not falling?
3. How do cyclists and motorcyclists keep balance?

Conclusions from the lesson:

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

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

Used sources and literature:

1. G. Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics. Grade 10.
2. Filmstrip "Stability" 1976 (scanned by me on a film scanner).
3. Disc "Movement and Interaction of Bodies" from "Electronic Lessons and Tests".
4. Disk "Work and Power" from "Electronic Lessons and Tests".

Obviously, the body can be at rest only in relation to one specific coordinate system. In statics, the conditions of equilibrium of bodies are studied in just such a system. In equilibrium, the speed and acceleration of all sections (elements) of the body are equal to zero. Taking this into account, one of the necessary conditions for the equilibrium of bodies can be established using the theorem on the motion of the center of mass (see § 7.4).

Internal forces do not affect the movement of the center of mass, since their sum is always zero. Only external forces determine the movement of the center of mass of a body (or a system of bodies). Since when the body is in equilibrium, the acceleration of all its elements is equal to zero, then the acceleration of the center of mass is also equal to zero. But the acceleration of the center of mass is determined by the vector sum of external forces applied to the body (see formula (7.4.2)). Therefore, in equilibrium, this amount should be zero.

Indeed, if the sum of external forces F i is equal to zero, then the acceleration of the center of mass is a c = 0. It follows that the speed of the center of mass is c = const. If at the initial moment the velocity of the center of mass was equal to zero, then the center of mass remains at rest in the future.

The obtained condition for the immobility of the center of mass is a necessary (but, as we will soon see, insufficient) condition for the equilibrium of a rigid body. This is the so-called first condition of equilibrium. It can be formulated as follows.

For the balance of the body, it is necessary that the sum of the external forces applied to the body be equal to zero:

If the sum of the forces is equal to zero, then the sum of the projections of the forces_on all three coordinate axes is also equal to zero. Denoting external forces through 1, 2, 3, etc., we obtain three equations equivalent to one vector equation (8.2.1):

In order for the body to rest, it is also necessary that the initial velocity of the center of mass be equal to zero.

The second condition for the equilibrium of a rigid body

Equality to zero of the sum of external forces acting on the body is necessary for balance, but not enough. When this condition is met, only the center of mass will necessarily be at rest. This is not difficult to be convinced of.

Let us apply forces equal in magnitude and opposite in direction to the board at different points, as shown in Figure 8.1 (two such forces are called a pair of forces). The sum of these forces is equal to zero: + (-) = 0. But the board will rotate. Only the center of mass is at rest if its initial velocity (velocity before the application of forces) was equal to zero.

Rice. 8.1

In the same way, two forces of the same magnitude and opposite in direction turn the handlebars of a bicycle or car (Fig. 8.2) around the axis of rotation.

Rice. 8.2

It is not difficult to understand what is the matter here. Any body is in equilibrium when the sum of all the forces acting on each of its elements is equal to zero. But if the sum of external forces is equal to zero, then the sum of all forces applied to each element of the body may not be equal to zero. In this case, the body will not be in balance. In the examples considered, the board and the steering wheel are therefore not in equilibrium because the sum of all the forces acting on the individual elements of these bodies is not equal to zero. The bodies are spinning.

Let us find out what other condition, besides the equality to zero of the sum of external forces, must be fulfilled so that the body does not rotate and is in equilibrium. For this we use the basic equation of the dynamics of the rotational motion of a rigid body (see § 7.6):

Recall that in formula (8.2.3)

is the sum of the moments of external forces applied to the body about the axis of rotation, and J is the moment of inertia of the body about the same axis.

If, then P = 0, that is, the body has no angular acceleration, and, therefore, the angular velocity of the body

If at the initial moment the angular velocity was equal to zero, then in the future the body will not perform rotational motion. Therefore, the equality

(at ω = 0) is the second condition necessary for the equilibrium of a rigid body.

When a rigid body is in equilibrium, the sum of the moments of all external forces acting on it with respect to any axis(1), is zero.

In the general case of an arbitrary number of external forces, the equilibrium conditions of a rigid body will be written in the form:

These conditions are necessary and sufficient for the equilibrium of any solid body. If they are fulfilled, then the vector sum of forces (external and internal) acting on each element of the body is equal to zero.

Equilibrium of deformable bodies

If the body is not absolutely solid, then under the action of external forces applied to it, it may not be in equilibrium, although the sum of the external forces and the sum of their moments relative to any axis is zero. This happens because under the action of external forces the body can deform and in the process of deformation the sum of all the forces acting on each of its elements, in this case, will not be equal to zero.

Let us apply, for example, to the ends of a rubber cord two forces equal in magnitude and directed along the cord in opposite directions. Under the action of these forces, the cord will not be in equilibrium (the cord is stretched), although the sum of the external forces is zero and the sum of their moments about the axis passing through any point of the cord is equal to zero.

When bodies are deformed, in addition, there is a change in the arms of forces and, consequently, a change in the moments of forces at given forces. Note also that only for rigid bodies it is possible to transfer the point of application of the force along the line of action of the force to any other point of the body. This does not change the moment of power and the internal state of the body.

In real bodies, it is possible to transfer the point of application of the force along the line of its action only when the deformations caused by this force are small and can be neglected. In this case, the change in the internal state of the body during the transfer of the point of application of the force is insignificant. If the deformations cannot be neglected, then such a transfer is unacceptable. So, for example, if two forces 1 and 2, equal in magnitude and directly opposite in direction, are applied along a rubber bar to its two ends (Fig. 8.3, a), then the bar will be stretched. When the points of application of these forces are transferred along the line of action to the opposite ends of the bar (Figure 8.3, b), the same forces will compress the bar and its internal state will be different.

Rice. 8.3

To calculate the equilibrium of deformable bodies, it is necessary to know their elastic properties, i.e., the dependence of deformations on the acting forces. We will not solve this difficult task. Simple cases of the behavior of deformable bodies will be considered in the next chapter.

(1) We considered the moments of forces relative to the real axis of rotation of the body. But it can be proved that when the body is in equilibrium, the sum of the moments of forces is equal to zero relative to any axis (geometric line), in particular, relative to three coordinate axes or relative to an axis passing through the center of mass.

The system of forces is called. balanced if under the action of this system the body remains at rest.

Equilibrium conditions:
The first condition for the equilibrium of a rigid body:
For the equilibrium of a rigid body, it is necessary that the sum of the external forces applied to the body be equal to zero.
The second condition for the equilibrium of a rigid body:
When a rigid body is in equilibrium, the sum of the moments of all external forces acting on it with respect to any axis is zero.
General condition of equilibrium of a rigid body:
For equilibrium of a rigid body, the sum of external forces and the sum of the moments of forces acting on the body must be equal to zero. The initial velocity of the center of mass and the angular velocity of rotation of the body must also be equal to zero.

Theorem. Three forces balance a rigid body only when they all lie in the same plane.

11. Flat system of forces Are forces located in the same plane.

Three forms of equilibrium equations for a flat system:

Center of gravity of the body.

Center of gravity a body of finite dimensions is a point relative to which the sum of the moments of gravity of all particles of the body is equal to zero. At this point, the body's gravity is applied. The center of gravity of a body (or a system of forces) usually coincides with the center of mass of a body (or a system of forces).

Center of gravity of a flat figure:

A practical way to find the center of mass of a flat figure: Suspend the body in a gravity field so that it can freely rotate around the suspension point O1 . In equilibrium, the center of mass WITH is on the same vertical line with the suspension point (below it), since it is equal to zero

moment of gravity, which can be considered applied at the center of mass. Changing the suspension point, in the same way we find another straight line О 2 С , passing through the center of mass. The position of the center of mass is given by the point of their intersection.

Center of Mass Velocity:

The momentum of a particle system is equal to the product of the mass of the entire system M = Σmi at the speed of its center of mass V :

The center of mass characterizes the movement of the system as a whole.

15. Sliding friction- friction during the relative motion of contacting bodies.

Rest friction- friction in the absence of relative movement of the contacting bodies.

Sliding friction force Ftr between the surfaces of contacting bodies during their relative motion depends on the force of the normal reaction N , or from the force of normal pressure Pn , and Ftr = kN or Ftr = kPn where k - coefficient of sliding friction depending on the same factors as the coefficient of static friction k0 , as well as on the speed of the relative motion of the contacting bodies.

16. Rolling friction Is the rolling of one body over another. The sliding friction force does not depend on the size of the rubbing surfaces, but only on the quality of the surfaces of the rubbing bodies and on the force that reduces the rubbing surfaces and is directed perpendicular to them. F = kN, where F- friction force, N- the magnitude of the normal reaction and k - sliding friction coefficient.

17. Equilibrium of bodies in the presence of friction is the maximum adhesion force proportional to the normal pressure of the body on the plane.

The angle between the total reaction, built on the greatest frictional force for a given normal reaction, and the direction of the normal reaction, is called angle of friction.

A cone with apex at the point of application of the normal reaction of a rough surface, the generatrix of which makes an angle of friction with this normal reaction, is called friction cone.

Dynamics.

1. V dynamics the influence of interactions between bodies on their mechanical motion is considered.

Weight is a painting characteristic of a material point. The mass is constant. The mass is adjetive (folds up)

Power - it is a vector that fully characterizes the interaction of a material point on it with other material points.

Material point- a body, the size and shape of which are insignificant in the considered motion. (Ex: in translational motion, a rigid body can be considered a material point)

System of material points called. many material points interacting with each other.

1 Newton's law: any material point maintains a state of rest or uniform rectilinear motion until external influences change this state.

2 Newton's law: the acceleration acquired by a material point in an inertial frame of reference is directly proportional to the force acting on the point, inversely proportional to the point's mass and coincides in direction with the force: a = F / m

Statics is a branch of mechanics that studies the conditions of equilibrium of bodies.

It follows from Newton's second law that if the geometric sum of all external forces applied to the body is zero, then the body is at rest or performs uniform rectilinear motion. In this case, it is customary to say that the forces applied to the body balance each other. When calculating resultant all forces acting on the body can be applied to center of mass .

For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero.

In fig. 1.14.1 gives an example of the equilibrium of a rigid body under the action of three forces. Intersection point O lines of action of forces and does not coincide with the point of application of the force of gravity (center of mass C), but in equilibrium these points are necessarily on the same vertical. When calculating the resultant, all forces are reduced to one point.

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

The rotating action of a force depends not only on its magnitude, but also on the distance between the line of action of the force and the axis of rotation.

The length of the perpendicular drawn from the axis of rotation to the line of action of the force is called shoulder of strength.

The product of the modulus of force on the shoulder d called moment of power M... The moments of those forces that tend to turn the body counterclockwise are considered positive (Fig. 1.14.2).

Rule of the Moments : a body with a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body relative to this axis is zero:

In the International System of Units (SI), the moments of forces are measured in Nnewton— meters (N ∙ m) .

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

A wheel rolling on a horizontal surface - an example indifferent equilibrium(fig. 1.14.3). If the wheel is stopped at any point, it will be in equilibrium. Along with indifferent equilibrium in mechanics, states sustainable and unstable balance.

A state of equilibrium is called stable if, with small deviations of the body from this state, forces or moments of forces arise that tend to return the body to an equilibrium state.

With a small deviation of the body from the state of unstable equilibrium, forces or moments of forces arise that tend to remove the body from the equilibrium position.

A ball lying on a flat horizontal surface is in a state of indifferent equilibrium. A ball at the top of a spherical protrusion is an example of an unstable balance. Finally, the ball at the bottom of the spherical depression is in a state of stable equilibrium (Figure 1.14.4).

For a body with a fixed axis of rotation, all three types of balance 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. Moreover, if the center of mass is below the axis of rotation, the state of equilibrium is stable. If the center of mass is located above the axis, the state of equilibrium is unstable (Fig. 1.14.5).

A special case is the balance of the body on the support. In this case, the elastic support force is applied not to one point, but is distributed over the base of the body. The body is in balance if a vertical line drawn through the center of mass of the body passes through support area, that is, inside the contour formed by the lines connecting the pivot points. If this line does not intersect the support area, then the body overturns. An interesting example of the balance of a body on a support is the leaning tower in the Italian city of Pisa (Fig. 1.14.6), 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 55 m high and a radius of 7 m. The top of the tower is deviated from the vertical by 4.5 m.

A vertical line drawn through the tower's center of gravity intersects the base at 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 happen very soon.