The equilibrium condition for bodies having an axis of rotation. Additional questions and tasks

A body is at rest (or moves uniformly and in a straight line) if the vector sum of all forces acting on it is zero. The forces are said to balance each other. When we are dealing with a body of a certain geometric shape, when calculating the resultant force, all forces can be applied to the center of mass of the body.

The condition for the equilibrium of bodies

In order for a body that does not rotate to be in equilibrium, it is necessary that the resultant of all forces acting on it be equal to zero.

F → = F 1 → + F 2 → + . . + F n → = 0 .

The figure above shows the equilibrium of a rigid body. The block is in a state of equilibrium under the action of three forces acting on it. The lines of action of the forces F 1 → and F 2 → intersect at the point O. The point of application of gravity is the center of mass of the body C. These points lie on one straight line, and when calculating the resultant force F 1 → , F 2 → and m g → are reduced to point C .

The condition that the resultant of all forces be equal to zero is not enough if the body can rotate around some axis.

The shoulder of the force d is the length of the perpendicular drawn from the line of action of the force to the point of its application. The moment of force M is the product of the arm of the force and its modulus.

The moment of force tends to rotate the body around its axis. Those moments that rotate the body counterclockwise are considered positive. The unit of measurement of the moment of force in the international SI system is 1 Newton meter.

Definition. moment rule

If the algebraic sum of all the moments applied to the body relative to the fixed axis of rotation is equal to zero, then the body is in equilibrium.

M1 + M2 + . . + M n = 0

Important!

In the general case, for the equilibrium of bodies, two conditions must be met: the resultant force is equal to zero and the rule of moments is observed.

There are different types of equilibrium in mechanics. Thus, a distinction is made between stable and unstable, as well as indifferent equilibrium.

A typical example of an indifferent equilibrium is a rolling wheel (or ball), which, if stopped at any point, will be in a state of equilibrium.

Stable equilibrium is such an equilibrium of a body when, with its small deviations, forces or moments of forces arise that tend to return the body to an equilibrium state.

Unstable equilibrium - a state of equilibrium, with a small deviation from which the forces and moments of forces tend to bring the body out of balance even more.

In the figure above, the position of the ball is (1) - indifferent equilibrium, (2) - unstable equilibrium, (3) - stable equilibrium.

A body with a fixed axis of rotation can be in any of the described equilibrium positions. If the axis of rotation passes through the center of mass, there is an indifferent equilibrium. In stable and unstable equilibrium, the center of mass is located on a vertical line that passes through the axis of rotation. When the center of mass is below the axis of rotation, the equilibrium is stable. Otherwise, vice versa.

A special case of equilibrium is the equilibrium of a body on a support. In this case, the elastic force is distributed over the entire base of the body, and does not pass through one point. A body is at rest in equilibrium when a vertical line drawn through the center of mass intersects the area of support. Otherwise, if the line from the center of mass does not fall into the contour formed by the lines connecting the support points, the body overturns.

An example of the balance of a body on a support is the famous Leaning Tower of Pisa. According to legend, Galileo Galilei dropped balls from it when he conducted his experiments on the study of the free fall of bodies.

A line drawn from the center of mass of the tower intersects the base approximately 2.3 m from its center.

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Definition

The equilibrium of the body is called such a state when any acceleration of the body is equal to zero, that is, all actions on the body of forces and moments of forces are balanced. In this case, the body can:

be in a state of calm;
move evenly and in a straight line;
rotate uniformly around an axis that passes through its center of gravity.

Body equilibrium conditions

If the body is in equilibrium, then two conditions are simultaneously satisfied.

The vector sum of all forces acting on the body is equal to the zero vector : $\sum_n((\overrightarrow(F))_n)=\overrightarrow(0)$
The algebraic sum of all moments of forces acting on the body is equal to zero: $\sum_n(M_n)=0$

The two equilibrium conditions are necessary but not sufficient. Let's take an example. Consider a wheel rolling uniformly without slipping on a horizontal surface. Both equilibrium conditions are met, but the body is moving.

Consider the case when the body does not rotate. In order for the body not to rotate and be in balance, it is necessary that the sum of the projections of all forces on an arbitrary axis be equal to zero, that is, the resultant of the forces. Then the body is either at rest, or moves uniformly and rectilinearly.

A body that has an axis of rotation will be in equilibrium if the rule of moments of forces is followed: the sum of the moments of forces that rotate the body clockwise must be equal to the sum of the moments of forces that rotate it counterclockwise.

To get the right moment with the least effort, you need to apply force as far as possible from the axis of rotation, increasing the same arm of the force and, accordingly, reducing the value of the force. Examples of bodies that have an axis of rotation are: a lever, doors, blocks, a brace, and the like.

Three types of balance of bodies that have a fulcrum

stable equilibrium, if the body, being removed from the equilibrium position to the neighboring nearest position and left in peace, returns to this position;
unstable equilibrium, if the body, being removed from the equilibrium position to a neighboring position and left at rest, will deviate even more from this position;
indifferent equilibrium - if the body, being brought to a neighboring position and left in peace, remains in its new position.

Balance of a body with a fixed axis of rotation

stable, if in the equilibrium position the center of gravity C occupies the lowest position of all possible near positions, and its potential energy will have the smallest value of all possible values in neighboring positions;
unstable if the center of gravity C occupies the highest of all nearby positions, and the potential energy has the greatest value;
indifferent if the center of gravity of the body C in all nearby possible positions is at the same level, and the potential energy does not change during the transition of the body.

Task 1

A body A with mass m = 8 kg is placed on a rough horizontal table surface. A thread is tied to the body, thrown over block B (Figure 1, a). What weight F can be tied to the end of the thread hanging from the block so as not to disturb the balance of the body A? Friction coefficient f = 0.4; ignore the friction on the block.

Let's define body weight ~A: ~G = mg = 8$\cdot $9.81 = 78.5 N.

We assume that all forces are applied to body A. When the body is placed on a horizontal surface, only two forces act on it: the weight G and the oppositely directed reaction of the support RA (Fig. 1, b).

If we apply some force F acting along a horizontal surface, then the reaction RA, which balances the forces G and F, will begin to deviate from the vertical, but the body A will be in equilibrium until the modulus of the force F exceeds the maximum value of the friction force Rf max , corresponding to the limit value of the angle $(\mathbf \varphi )$o (Fig. 1, c).

Having decomposed the reaction RA into two components Rf max and Rn, we obtain a system of four forces applied to one point (Fig. 1, d). Projecting this system of forces onto the x and y axes, we obtain two equilibrium equations:

$(\mathbf \Sigma )Fkx = 0, F - Rf max = 0$;

$(\mathbf \Sigma )Fky = 0, Rn - G = 0$.

We solve the resulting system of equations: F = Rf max, but Rf max = f$\cdot $ Rn, and Rn = G, so F = f$\cdot $ G = 0.4$\cdot $ 78.5 = 31.4 H; m \u003d F / g \u003d 31.4 / 9.81 \u003d 3.2 kg.

Answer: Mass of cargo m = 3.2 kg

Task 2

The system of bodies shown in Fig. 2 is in a state of equilibrium. Weight of cargo tg=6 kg. Angle between vectors $\widehat((\overrightarrow(F))_1(\overrightarrow(F))_2)=60()^\circ $. $\left|(\overrightarrow(F))_1\right|=\left|(\overrightarrow(F))_2\right|=F$. Find the mass of weights.

The resultant force $(\overrightarrow(F))_1and\ (\overrightarrow(F))_2$ is equal in absolute value to the weight of the load and opposite to it in direction: $\overrightarrow(R)=(\overrightarrow(F))_1+(\overrightarrow (F))_2=\ -m\overrightarrow(g)$. By the law of cosines, $(\left|\overrightarrow(R)\right|)^2=(\left|(\overrightarrow(F))_1\right|)^2+(\left|(\overrightarrow(F) )_2\right|)^2+2\left|(\overrightarrow(F))_1\right|\left|(\overrightarrow(F))_2\right|(cos \widehat((\overrightarrow(F)) _1(\overrightarrow(F))_2)\ )$.

Hence $(\left(mg\right))^2=$; $F=\frac(mg)(\sqrt(2\left(1+(cos 60()^\circ \ )\right)))$;

Since the blocks are movable, $m_g=\frac(2F)(g)=\frac(2m)(\sqrt(2\left(1+\frac(1)(2)\right)))=\frac(2 \cdot 6)(\sqrt(3))=6.93\ kg\ $

Answer: The mass of each weight is 6.93 kg.

Lesson objectives:

Educational. To study two conditions for the equilibrium of bodies, types of equilibrium (stable, unstable, indifferent). Find out under what conditions bodies are more stable.

Developing: To promote the development of cognitive interest in physics, to develop the ability to make comparisons, generalize, highlight the main thing, draw conclusions.

Educational: to cultivate discipline, attention, the ability to express their point of view and defend it.

Lesson plan:

1. Knowledge update

2. What is static

3. What is balance. Types of balance

4. Center of gravity

5. Problem solving

Lesson progress:

1.Updating knowledge.

Teacher: Hello!

Students: Hello!

Teacher: We continue to talk about forces. In front of you is an irregularly shaped body (stone), suspended on a thread and attached to an inclined plane. What forces are acting on this body?

Students: The body is affected by: the tension force of the thread, the force of gravity, the force tending to tear off the stone, opposite to the tension force of the thread, the reaction force of the support.

Teacher: Forces found, what do we do next?

Students: Write down Newton's second law.

There is no acceleration, so the sum of all forces is zero.

Teacher: What does it say?

Students: This indicates that the body is at rest.

Teacher: Or you can say that the body is in a state of equilibrium. The equilibrium of a body is the state of rest of that body. Today we will talk about the balance of bodies. Write down the topic of the lesson: "Equilibrium conditions for bodies. Types of equilibrium."

2. Formation of new knowledge and methods of action.

Teacher: The section of mechanics that studies the equilibrium of absolutely rigid bodies is called statics. There is not a single body around us that would not be affected by forces. Under the influence of these forces, the bodies are deformed.

When elucidating the equilibrium conditions for deformed bodies, it is necessary to take into account the magnitude and nature of the deformation, which complicates the task put forward. Therefore, in order to clarify the basic laws of equilibrium, for convenience, the concept of an absolutely rigid body was introduced.

An absolutely rigid body is a body in which the deformations that occur under the action of forces applied to it are negligible. Write down the definitions of statics, balance of bodies and absolutely rigid body from the screen (slide 2).

And the fact that we have found out that the body is in equilibrium if the geometric sum of all the forces applied to it is zero is the first condition for equilibrium. Write down 1 equilibrium condition:

If the sum of the forces is equal to zero, then the sum of the projections of these forces on the coordinate axes is also equal to zero. In particular, for the projections of external forces on the X axis, we can write .

Equality to zero of the sum of external forces acting on a rigid body is necessary for its equilibrium, but not enough. For example, two equal and oppositely directed forces were applied to the board at different points. The sum of these forces is zero. Will the board be in balance?

Students: The board will turn, for example, like the steering wheel of a bicycle or car.

Teacher: Right. In the same way, two identical in magnitude and oppositely directed forces turn the steering wheel of a bicycle or car. Why is this happening?

Students: ???

Teacher: Any body is in equilibrium when the sum of all 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 equilibrium. Therefore, we need to find out one more condition for the equilibrium of bodies. To do this, we will conduct an experiment. (Two students are called.) One of the students applies force closer to the axis of rotation of the door, the other student - closer to the handle. They apply forces in different directions. What happened?

Students: The one that applied the force closer to the handle won.

Teacher: Where is the line of action of the force applied by the first disciple?

Students: Closer to the axis of rotation of the door.

Teacher: Where is the line of action of the force applied by the second student?

Students: Closer to the doorknob.

Teacher: What else can we notice?

Students: That the distances from the axis of rotation to the lines of application of forces are different.

Teacher: So what else determines the result of the action of force?

Students: The result of the action of the force depends on the distance from the axis of rotation to the line of action of the force.

Teacher: What is the distance from the axis of rotation to the line of action of the force?

Students: Shoulder. The shoulder is a perpendicular drawn from the axis of rotation to the line of action of this force.

Teacher: How do forces and shoulders relate to each other in this case?

Students: According to the rule of equilibrium of a lever, the forces acting on it are inversely proportional to the shoulders of these forces. .

Teacher: What is the product of the modulus of the force that rotates the body and its arm?

Students: Moment of power.

Teacher: So the moment of force applied to the first students is , and the moment of force applied to the second students is

Now we can formulate the second equilibrium condition: A solid body is in equilibrium if the algebraic sum of the moments of external forces acting on it about any axis is zero. (Slide 3)

Let us introduce the concept of the center of gravity. The center of gravity is the point of application of the resultant force of gravity (the point through which the resultant of all parallel gravity forces acting on individual elements of the body passes). There is also the concept of the center of mass.

The center of mass of a system of material points is called a geometric point, the coordinates of which are determined by the formula:

; same for .

The center of gravity coincides with the center of mass of the system if this system is in a uniform gravitational field.

Look at the screen. Try to find the center of gravity of these figures. (slide 4)

(Demonstrate with the help of a bar with recesses and slides and a ball types of balance.)

On slide 5 you see what you saw in experience. Write down the equilibrium stability conditions from slides 6,7,8:

1. Bodies are in a state of stable equilibrium if, at the slightest deviation from the equilibrium position, a force or moment of force arises that returns the body to the equilibrium position.

2. Bodies are in a state of unstable equilibrium if, at the slightest deviation from the equilibrium position, a force or moment of force arises that removes the body from the equilibrium position.

3. Bodies are in a state of indifferent equilibrium if, at the slightest deviation from the equilibrium position, neither a force nor a moment of force arises that changes the position of the body.

Now look at slide 9. What can you say about the stability conditions in all three cases.

Students: In the first case, if the fulcrum is higher than the center of gravity, then the equilibrium is stable.

In the second case, if the fulcrum coincides with the center of gravity, then the equilibrium is indifferent.

In the third case, if the center of gravity is higher than the fulcrum, the balance is unstable.

Teacher: Now let's consider bodies that have a support area. The area of support is understood as the area of contact of the body with the support. (slide 10).

Let us consider how the position of the line of action of the force of gravity changes with respect to the axis of rotation of the body when the body with the area of support is tilted. (slide 11)

Note that as the body rotates, the position of the center of gravity changes. And any system always tends to lower the position of the center of gravity. So inclined bodies will be in a state of stable equilibrium, while the line of action of gravity will pass through the area of support. Look at slide 12.

If the deflection of a body having an area of support increases the center of gravity, then the balance will be stable. In stable equilibrium, a vertical line passing through the center of gravity will always pass through the area of support.

Two bodies that have the same weight and area of support, but different heights, have different limiting angles of inclination. If this angle is exceeded, then the bodies overturn. (slide 13)

With a lower center of gravity, more work must be expended to tip the body. Therefore, the work of overturning can serve as a measure of its stability. (Slide 14)

So inclined structures are in a position of stable equilibrium, because the line of action of gravity passes through the area of their support. For example, the Leaning Tower of Pisa.

The swaying or tilting of the human body when walking is also explained by the desire to maintain a stable position. The support area is determined by the area inside the line drawn around the extreme points of contact with the support body. when the person is standing. The line of action of gravity passes through the support. When a person lifts his leg, in order to maintain balance, he bends over, transferring the line of action of gravity to a new position so that it again passes through the area of support. (slide 15)

For the stability of various structures, the support area is increased or the center of gravity of the structure is lowered, making a powerful support, or the support area is increased and, at the same time, the center of gravity of the structure is lowered.

The stability of transport is determined by the same conditions. So, of the two modes of transport, a car and a bus, a car is more stable on an inclined road.

With the same inclination of these modes of transport near the bus, the line of gravity runs closer to the edge of the support area.

Problem solving

Task: Material points with masses m, 2m, 3m and 4m are located at the vertices of a rectangle with sides 0.4m and 0.8m. Find the center of gravity of the system of these material points.

x s -? at with -?

Finding the center of gravity of a system of material points means finding its coordinates in the XOY coordinate system. Let us align the origin of coordinates XOY with the vertex of the rectangle containing the material point of mass m, and direct the coordinate axes along the sides of the rectangle. The coordinates of the center of gravity of the system of material points are equal to:

Here, is the coordinate on the OX axis of a point with mass . As follows from the drawing, because this point is located at the origin. The coordinate is also equal to zero, the coordinates of points with masses on the OX axis are the same and equal to the length of the side of the rectangle. Substituting the values of the coordinates, we get

The coordinate on the OY axis of a point with mass is zero, =0. The coordinates of points with masses on this axis are the same and equal to the length of the side of the rectangle. Substituting these values, we get

Test questions:

1. Conditions for the equilibrium of the body?

1 equilibrium condition:

A rigid body is in equilibrium if the geometric sum of the external forces applied to it is zero.

2 Equilibrium condition: A solid body is in equilibrium if the algebraic sum of the moments of external forces acting on it about any axis is equal to zero.

2. Name the types of balance.

Bodies are in a state of stable equilibrium if, at the slightest deviation from the equilibrium position, a force or moment of force arises that returns the body to the equilibrium position.

Bodies are in a state of unstable equilibrium if, at the slightest deviation from the equilibrium position, a force or moment of force arises that removes the body from the equilibrium position.

Bodies are in a state of indifferent equilibrium if, at the slightest deviation from the equilibrium position, neither a force nor a moment of force arises that changes the position of the body.

Homework:

List of used literature:

1. Physics. Grade 10: textbook. for general education institutions: basic and profile. levels / G. Ya. Myakishev, B. B. Bukhovtsev, N. N. Sotsky; ed. V. I. Nikolaev, N. A. Parfenteva. - 19th ed. - M.: Enlightenment, 2010. - 366 p.: ill.
2. Maron A.E., Maron E.A. "Collection of qualitative problems in physics 10 cells, M.: Enlightenment, 2006
3. L.A. Kirik, L.E.Gendenshtein, Yu.I.Dik. Methodical materials for the teacher grade 10, M.: Ileksa, 2005.-304s:, 2005
4. L.E.Gendenshtein, Yu.I.Dik. Physics grade 10.-M.: Mnemosyne, 2010

In physics for grade 9 (I.K. Kikoin, A.K. Kikoin, 1999),
task №6
to chapter " LABORATORY WORKS».

The purpose of the work: to establish the ratio between the moments of forces applied to the arms of the lever when it is in equilibrium. To do this, one or more weights are suspended from one of the arms of the lever, and a dynamometer is attached to the other (Fig. 179).

This dynamometer measures the modulus of force F, which must be applied in order for the lever to be in balance. Then, with the help of the same dynamometer, the modulus of the weight of the goods P is measured. The lever arm lengths are measured with a ruler. After that, the absolute values of the moments M 1 and M 2 of the forces P and F are determined:

The conclusion about the error of the experimental verification of the moment rule can be made by comparing with unity

relation:

Measuring:

1) ruler; 2) dynamometer.

Materials: 1) tripod with clutch; 2) lever; 3) a set of goods.

Work order

1. Mount the arm on a tripod and balance it in a horizontal position using the sliding nuts located at its ends.

2. Hang a load at some point on one of the arms of the lever.

3. Attach a dynamometer to the other arm of the lever and determine the force to be applied.

live towards the lever so that it is in balance.

4. Use a ruler to measure the length of the lever arms.

5. Using a dynamometer, determine the weight of the load R.

6. Find the absolute values of the moments of forces P and F

7. Enter the found values in the table:

				M 1 \u003d Pl 1, N⋅m

8. Compare the ratio

with unity and draw a conclusion about the error of the experimental verification of the moment rule.

The main purpose of the work is to establish the relationship between the moments of forces applied to a body with a fixed axis of rotation at its equilibrium. In our case, we use a lever as such a body. According to the rule of moments, for such a body to be in equilibrium, it is necessary that the algebraic sum of the moments of forces about the axis of rotation be equal to zero.

Consider such a body (in our case, a lever). Two forces act on it: the weight of the loads P and the force F (the elasticity of the spring of the dynamometer), so that the lever is in balance and the moments of these forces must be equal in absolute value to each other. The absolute values of the moments of forces F and P will be determined respectively:

Conclusions about the error of the experimental verification of the moment rule can be made by comparing the ratio with unity:

Measuring instruments: ruler (Δl = ±0.0005 m), dynamometer (ΔF = ±0.05 H). The mass of weights from the set in mechanics is assumed to be (0.1 ± 0.002) kg.

Completing of the work

Let the body be fixed on a fixed axis (item 1.4) and a force is applied to it in one of two ways:

1) the line of action passes through the axis of rotation. will be balanced by the reaction and the body is in balance;

2) the line of action does not pass through the axis of rotation, which causes the body to rotate.

Let us apply a force to the body that causes it to rotate in the opposite direction. Under certain conditions, the rotation can become uniform or stop altogether. It is known from experiments that this will happen if , where d 1 and d 2 – shoulders strength and .

Shoulder of Strength(d)about the axis- the shortest distance from the line of action of the force to this axis.

Moment of power (M) is the product of the force modulus and its shoulder.

[M] = 1 Nm

· In this paragraph, the moment is considered as a scalar quantity, and the forces and their arms lie in a plane perpendicular to the axis of rotation.

The moment of force rotating the body clockwise is considered negative, counter-clockwise is considered positive.

The equilibrium condition is known as moment rule: a body with a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to it is zero.

Complete equilibrium condition (for any bodies)

A body is in equilibrium if the resultant of all forces applied to it is zero and the sum of the moments of these forces about the axis of rotation is also zero.

Types of balance

1. sustainable balance- equilibrium, upon exit from which a force appears, returning the body to its original position.

2. Unstable equilibrium- equilibrium, upon exit from which a force arises, which further deviates the body from its original position.

3. Indifferent balance- equilibrium, upon exit from which neither a restoring nor a deflecting force arises.

MOLECULAR PHYSICS

Molecular physics- a branch of physics in which the phenomena of changing the state of bodies and substances are explained from the point of view of the internal structure of matter.

Origins of molecular physics

Representations of the ancients

Ancient philosophical schools explained the structure of bodies and substances in different ways. For example, in China, scientists believed that bodies consist of water, fire, ether, air, etc. Leucippus (5th century BC, Greece) and Democritus (5th century BC, Greece) expressed the idea that:

1) all bodies consist of the smallest particles - atoms;

2) differences between bodies are determined either by the difference in their atoms, or by the difference in the arrangement of atoms.

Development of molecular physics

Mikhail Vasilyevich Lomonosov (1711–1765, Russia) made a great contribution to science. He developed the idea of the molecular (atomic) structure of matter and suggested that:

1) particles (molecules) move randomly;

2) the speed of movement of molecules is related to the temperature of the substance (the higher the temperature, the higher the speed);

3) there must be a temperature at which the movement of molecules stops.

Experiments carried out in the 19th century confirmed the correctness of his ideas.

Brown's experience

In 1827, the botanist Robert Brown (1773–1858, England) placed a liquid with fine solids in it under a microscope and found that:

1) particles move randomly;

2) the smaller the particle, the stronger its movement is noticeable;

He came to the conclusion that the impacts of solid particles are given by liquid particles during collisions. The works of many scientists developed the doctrine of the structure and properties of matter - the molecular-kinetic theory (MKT), based on the concept of the existence of molecules (atoms).

Basic provisions of the ICB

1) Substances consist of particles: atoms and molecules;

2) particles move randomly;

3) particles interact with each other.

Based on these provisions, the following phenomena were explained: the elasticity of gases, liquids and solids; transfer of matter from one state of aggregation to another; expansion of gases; diffusion and etc.

Aggregate state (thermodynamic phase)- one of the three states of matter (solid, liquid, gaseous).

Diffusion- spontaneous mixing of substances.