Designation of the moment. Statics

In this lesson, the topic of which is “Moment of Force,” we will talk about the force that must be applied to a body in order to change its speed, as well as the point of application of this force. Let's look at examples of rotation of different bodies, for example a swing: at what point should a force be applied in order for the swing to start moving or remain in balance.

Imagine that you are a football player and there is a soccer ball in front of you. To make it fly, you need to hit it. It’s simple: the harder you hit, the faster and further it will fly, and you’ll most likely hit the center of the ball (see Fig. 1).

And in order for the ball to rotate in flight and fly along a curved trajectory, you will not hit the center of the ball, but from the side, which is what football players do to deceive their opponents (see Fig. 2).

Rice. 2. Curved trajectory of the ball

Here it is already important which point to hit.

Another simple question: in what place should you take the stick so that it does not tip over when lifting? If the stick is uniform in thickness and density, then we will take it in the middle. What if it is more massive on one end? Then we will take it closer to the massive edge, otherwise it will outweigh (see Fig. 3).

Rice. 3. Lifting point

Imagine: dad sat on a balance swing (see Fig. 4).

Rice. 4. Balance swing

To outweigh it, you will sit on the swing closer to the opposite end.

In all the examples given, it was important for us not only to act on the body with some force, but it was also important in what place, on what point of the body to act. We chose this point at random, using life experience. What if there are three different weights on the stick? What if you lift it together? What if we are talking about a crane or a cable-stayed bridge (see Fig. 5)?

Rice. 5. Examples from life

To solve such problems, intuition and experience are not enough. Without a clear theory, they can no longer be solved. Today we will talk about solving such problems.

Usually in problems we have a body to which forces are applied, and we solve them, as always before, without thinking about the point of application of the force. It is enough to know that the force is applied simply to the body. Such problems occur often, we know how to solve them, but it happens that it is not enough to simply apply force to the body - it becomes important at what point.

An example of a problem in which body size is not important

For example, there is a small iron ball on the table, which is subject to a gravitational force of 1 N. What force must be applied to lift it? The ball is attracted by the Earth, we will act upward on it, applying some force.

The forces acting on the ball are directed in opposite directions, and in order to lift the ball, you need to act on it with a force greater in magnitude than the force of gravity (see Fig. 6).

Rice. 6. Forces acting on the ball

The force of gravity is equal to , which means that the ball needs to be acted upward with a force:

We didn’t think about how exactly we take the ball, we just take it and lift it. When we show how we lifted the ball, we can easily draw a dot and show: we acted on the ball (see Fig. 7).

Rice. 7. Action on the ball

When we can do this with a body, show it in a drawing when explaining it in the form of a point and not pay attention to its size and shape, we consider it a material point. This is a model. In reality, the ball has a shape and dimensions, but we did not pay attention to them in this problem. If the same ball needs to be made to rotate, then it is no longer possible to simply say that we are influencing the ball. The important thing here is that we pushed the ball from the edge and not into the center, causing it to rotate. In this problem, the same ball can no longer be considered a point.

We already know examples of problems in which we need to take into account the point of application of force: a problem with a soccer ball, with a non-uniform stick, with a swing.

The point of application of force is also important in the case of a lever. Using a shovel, we act on the end of the handle. Then it is enough to apply a small force (see Fig. 8).

Rice. 8. Low force action on the shovel handle

What do the considered examples have in common, where it is important for us to take into account body size? And the ball, and the stick, and the swing, and the shovel - in all these cases we were talking about the rotation of these bodies around a certain axis. The ball rotated around its axis, the swing rotated around the mount, the stick around the place in which we held it, the shovel around the fulcrum (see Fig. 9).

Rice. 9. Examples of rotating bodies

Let's consider the rotation of bodies around a fixed axis and see what makes the body rotate. We will consider rotation in one plane, then we can assume that the body rotates around one point O (see Fig. 10).

Rice. 10. Pivot point

If we want to balance a swing whose beam is glass and thin, then it may simply break, and if the beam is made of soft metal and also thin, it may bend (see Fig. 11).

We will not consider such cases; We will consider the rotation of strong rigid bodies.

It would be incorrect to say that rotational motion is determined only by force. After all, on a swing, the same force can cause it to rotate, or it may not, depending on where we sit. It's not just a matter of strength, but also the location of the point on which we act. Everyone knows how difficult it is to lift and hold a load at arm's length. To determine the point of application of force, the concept of the shoulder of force is introduced (by analogy with the shoulder of the hand with which a load is lifted).

The arm of a force is the minimum distance from a given point to the straight line along which the force acts.

From geometry you probably already know that this is a perpendicular dropped from point O to a straight line along which the force acts (see Fig. 12).

Rice. 12. Graphic representation of the leverage

Why is the arm of a force the minimum distance from point O to the straight line along which the force acts?

It may seem strange that the arm of a force is measured from point O not to the point of application of the force, but to the straight line along which this force acts.

Let's do the following experiment: tie a thread to the lever. Let's act on the lever with some force at the point where the thread is tied (see Fig. 13).

Rice. 13. The thread is tied to the lever

If enough torque is created to turn the lever, it will turn. The thread will show a straight line along which the force is directed (see Fig. 14).

Let's try to pull the lever with the same force, but now holding the thread. Nothing will change in the effect on the lever, although the point of application of the force will change. But the force will act along the same straight line, its distance to the axis of rotation, that is, the arm of the force, will remain the same. Let's try to operate the lever at an angle (see Fig. 15).

Rice. 15. Action on the lever at an angle

Now the force is applied to the same point, but acts along a different line. Its distance to the axis of rotation has become small, the moment of force has decreased, and the lever may no longer turn.

The body is subjected to an influence aimed at rotation, at turning the body. This impact depends on the force and its leverage. The quantity characterizing the rotational effect of force on a body is called moment of power, sometimes also called torque or torque.

The meaning of the word "moment"

We are accustomed to using the word “moment” to mean a very short period of time, as a synonym for the word “moment” or “moment.” Then it is not entirely clear what relation the moment has to force. Let us turn to the origin of the word “moment”.

The word comes from the Latin momentum, which means “driving force, push.” The Latin verb movēre means “to move” (as does the English word move, and movement means “movement”). Now it is clear to us that torque is what makes a body rotate.

The moment of a force is the product of the force and its arm.

The unit of measurement is newton multiplied by meter: .

If you increase the force arm, you can decrease the force and the moment of force will remain the same. We use this very often in everyday life: when we open a door, when we use pliers or a wrench.

The last point of our model remains - we need to figure out what to do if several forces act on the body. We can calculate the moment of each force. It is clear that if the forces rotate the body in one direction, then their action will add up (see Fig. 16).

Rice. 16. The action of forces adds up

If in different directions, the moments of force will balance each other and it is logical that they will need to be subtracted. Therefore, we will write the moments of forces that rotate the body in different directions with different signs. For example, let's write down if the force supposedly rotates the body around an axis clockwise, and if it rotates counterclockwise (see Fig. 17).

Rice. 17. Definition of signs

Then we can write down one important thing: for a body to be in equilibrium, the sum of the moments of the forces acting on it must be equal to zero.

Formula for leverage

We already know the principle of operation of a lever: two forces act on the lever, and the greater the lever arm, the lesser the force:

Let's consider the moments of forces that act on the lever.

Let's choose a positive direction of rotation of the lever, for example counterclockwise (see Fig. 18).

Rice. 18. Selecting the direction of rotation

Then the moment of force will have a plus sign, and the moment of force will have a minus sign. For the lever to be in equilibrium, the sum of the moments of forces must be equal to zero. Let's write down:

Mathematically, this equality and the relationship written above for the lever are one and the same, and what we obtained experimentally was confirmed.

For example, Let's determine whether the lever shown in the figure will be in equilibrium. Three forces act on it(see Fig. 19) . , And. Shoulders of forces are equal, And.

Rice. 19. Drawing for problem 1

For the lever to be in equilibrium, the sum of the moments of forces that act on it must be equal to zero.

According to the condition, three forces act on the lever: , and . Their shoulders are respectively equal to , and .

The direction of rotation of the lever clockwise will be considered positive. In this direction the lever is rotated by a force, its moment is equal to:

The forces and rotate the lever counterclockwise, we write their moments with a minus sign:

It remains to calculate the sum of the moments of forces:

The total moment is not equal to zero, which means that the body will not be in equilibrium. The total moment is positive, which means the lever will rotate clockwise (in our problem this is the positive direction).

We solved the problem and got the result: the total moment of forces acting on the lever is equal to . The lever will begin to turn. And when it turns, if the forces do not change direction, the shoulders of the forces will change. They will decrease until they become zero when the lever is turned vertical (see Fig. 20).

Rice. 20. Shoulder forces are zero

And with further rotation, the forces will become directed so as to rotate it in the opposite direction. Therefore, having solved the problem, we determined in which direction the lever would begin to rotate, not to mention what would happen next.

Now you have learned to determine not only the force with which you need to act on the body in order to change its speed, but also the point of application of this force so that it does not turn (or turn, as we need).

How to push a cabinet without it tipping over?

We know that when we push a cabinet with force at the top, it will tip over, and to prevent this from happening, we push it lower. Now we can explain this phenomenon. The axis of its rotation is located on the edge on which it stands, while the shoulders of all forces, except for the force, are either small or equal to zero, therefore, under the influence of force, the cabinet falls (see Fig. 21).

Rice. 21. Action on the top of the cabinet

By applying a force below, we reduce its shoulder, which means the moment of this force and overturning does not occur (see Fig. 22).

Rice. 22. Force applied below

The cabinet as a body, the dimensions of which we take into account, obeys the same law as a wrench, a door handle, bridges on supports, etc.

This concludes our lesson. Thank you for your attention!

Bibliography

Sokolovich Yu.A., Bogdanova G.S. Physics: A reference book with examples of problem solving. - 2nd edition repartition. - X.: Vesta: Ranok Publishing House, 2005. - 464 p.
Peryshkin A.V. Physics. 7th grade: textbook. for general education institutions - 10th ed., add. - M.: Bustard, 2006. - 192 p.: ill.

Abitura.com ().
solverbook.com ().

Homework

The rule of leverage, discovered by Archimedes in the third century BC, existed for almost two thousand years, until in the seventeenth century, with the light hand of the French scientist Varignon, it received a more general form.

Torque rule

The concept of torque was introduced. The moment of force is a physical quantity equal to the product of the force and its arm:

where M is the moment of force,
F - strength,
l - leverage of force.

From the lever equilibrium rule directly The rule for moments of forces follows:

F1 / F2 = l2 / l1 or, by the property of proportion, F1 * l1= F2 * l2, that is, M1 = M2

In verbal expression, the rule of moments of forces is as follows: a lever is in equilibrium under the action of two forces if the moment of the force rotating it clockwise is equal to the moment of the force rotating it counterclockwise. The rule of moments of force is valid for any body fixed around a fixed axis. In practice, the moment of force is found as follows: in the direction of action of the force, a line of action of the force is drawn. Then, from the point at which the axis of rotation is located, a perpendicular is drawn to the line of action of the force. The length of this perpendicular will be equal to the arm of the force. By multiplying the value of the force modulus by its arm, we obtain the value of the moment of force relative to the axis of rotation. That is, we see that the moment of force characterizes the rotating action of the force. The effect of a force depends on both the force itself and its leverage.

Application of the rule of moments of forces in various situations

This implies the application of the rule of moments of forces in various situations. For example, if we open a door, then we will push it in the area of the handle, that is, away from the hinges. You can do a basic experiment and make sure that pushing the door is easier the further we apply force from the axis of rotation. The practical experiment in this case is directly confirmed by the formula. Since, in order for the moments of forces at different arms to be equal, it is necessary that the larger arm correspond to a smaller force and, conversely, the smaller arm correspond to a larger one. The closer to the axis of rotation we apply the force, the greater it should be. The farther from the axis we operate the lever, rotating the body, the less force we will need to apply. Numerical values can be easily found from the formula for the moment rule.

It is precisely based on the rule of moments of force that we take a crowbar or a long stick if we need to lift something heavy, and, having slipped one end under the load, we pull the crowbar near the other end. For the same reason, we screw in the screws with a long-handled screwdriver, and tighten the nuts with a long wrench.

A moment of power relative to an arbitrary center in the plane of action of the force, the product of the force modulus and the shoulder is called.

Shoulder- the shortest distance from the center O to the line of action of the force, but not to the point of application of the force, because force-sliding vector.

Moment sign:

Clockwise - minus, counterclockwise - plus;

The moment of force can be expressed as a vector. This is perpendicular to the plane according to Gimlet's rule.

If several forces or a system of forces are located in the plane, then the algebraic sum of their moments will give us main point systems of forces.

Let's consider the moment of force about the axis, calculate the moment of force about the Z axis;

Let's project F onto XY;

F xy =F cosα= ab

m 0 (F xy)=m z (F), that is, m z =F xy * h= F cosα* h

The moment of force relative to the axis is equal to the moment of its projection onto the plane perpendicular to the axis, taken at the intersection of the axes and the plane

If the force is parallel to the axis or intersects it, then m z (F)=0

Expressing moment of force as a vector expression

Let's draw r a to point A. Consider OA x F.

This is the third vector m o , perpendicular to the plane. The magnitude of the cross product can be calculated using twice the area of the shaded triangle.

Analytical expression of force relative to coordinate axes.

Let us assume that the Y and Z, X axes with unit vectors i, j, k are associated with point O. Considering that:

r x =X * Fx ; r y =Y * F y ; r z =Z * F y we get: m o (F)=x =

Let's expand the determinant and get:

m x =YF z - ZF y

m y =ZF x - XF z

m z =XF y - YF x

These formulas make it possible to calculate the projection of the vector moment on the axis, and then the vector moment itself.

Varignon's theorem on the moment of the resultant

If a system of forces has a resultant, then its moment relative to any center is equal to the algebraic sum of the moments of all forces relative to this point

If we apply Q= -R, then the system (Q,F 1 ... F n) will be equally balanced.

The sum of the moments about any center will be equal to zero.

Analytical equilibrium condition for a plane system of forces

This is a flat system of forces, the lines of action of which are located in the same plane

The purpose of calculating problems of this type is to determine the reactions of external connections. To do this, the basic equations in a plane system of forces are used.

2 or 3 moment equations can be used.

Example

Let's create an equation for the sum of all forces on the X and Y axis:

The sum of the moments of all forces relative to point A:

Parallel forces

Equation for point A:

Equation for point B:

The sum of the projections of forces on the Y axis.

The moment of a force relative to an axis, or simply the moment of force, is the projection of a force onto a straight line, which is perpendicular to the radius and drawn at the point of application of the force, multiplied by the distance from this point to the axis. Or the product of the force and the shoulder of its application. The shoulder in this case is the distance from the axis to the point of application of force. The moment of force characterizes the rotational action of a force on a body. The axis in this case is the place where the body is attached, about which it can rotate. If the body is not fixed, then the axis of rotation can be considered the center of mass.

Formula 1 - Moment of force.

F - Force acting on the body.

r - Leverage of force.

Figure 1 - Moment of force.

As can be seen from the figure, the force arm is the distance from the axis to the point of application of the force. But this is if the angle between them is 90 degrees. If this is not the case, then it is necessary to draw a line along the action of the force and lower a perpendicular from the axis onto it. The length of this perpendicular will be equal to the arm of the force. But moving the point of application of a force along the direction of the force does not change its moment.

It is generally accepted that a moment of force that causes a body to rotate clockwise relative to the observation point is considered positive. And negative, respectively, causing rotation against it. The moment of force is measured in Newtons per meter. One Newtonometer is a force of 1 Newton acting on an arm of 1 meter.

If the force acting on the body passes along a line running through the axis of rotation of the body, or the center of mass, if the body does not have an axis of rotation. Then the moment of force in this case will be equal to zero. Since this force will not cause rotation of the body, but will simply move it translationally along the line of application.

Figure 2 - The moment of force is zero.

If several forces act on a body, then the moment of force will be determined by their resultant. For example, two forces of equal magnitude and opposite directions can act on a body. In this case, the total moment of force will be equal to zero. Since these forces will compensate each other. To put it simply, imagine a children's carousel. If one boy pushes it clockwise, and the other with the same force against it, then the carousel will remain motionless.

Definition

The vector product of the radius - vector (), which is drawn from point O (Fig. 1) to the point to which the force is applied to the vector itself is called the moment of force () with respect to point O:

In Fig. 1, point O and the force vector () and radius vector are in the plane of the figure. In this case, the vector of the moment of force () is perpendicular to the plane of the drawing and has a direction away from us. The vector of the moment of force is axial. The direction of the force moment vector is chosen in such a way that rotation around point O in the direction of force and the vector create a right-handed system. The direction of the moment of forces and angular acceleration coincide.

The magnitude of the vector is:

where is the angle between the radius and force vector directions, is the force arm relative to point O.

Moment of force about the axis

The moment of force relative to an axis is a physical quantity equal to the projection of the vector of the moment of force relative to the point of the chosen axis onto a given axis. In this case, the choice of point does not matter.

The main moment of strength

The main moment of a set of forces relative to point O is called a vector (moment of force), which is equal to the sum of the moments of all forces acting in the system in relation to the same point:

In this case, point O is called the center of reduction of the system of forces.

If there are two main moments ( and ) for one system of forces for different two centers of bringing forces (O and O’), then they are related by the expression:

where is the radius vector, which is drawn from point O to point O’, is the main vector of the force system.

In the general case, the result of the action of an arbitrary system of forces on a rigid body is the same as the action on the body of the main moment of the system of forces and the main vector of the system of forces, which is applied at the center of reduction (point O).

Basic law of the dynamics of rotational motion

where is the angular momentum of a body in rotation.

For a solid body this law can be represented as:

where I is the moment of inertia of the body, and is the angular acceleration.

Torque units

The basic unit of measurement of moment of force in the SI system is: [M]=N m

In GHS: [M]=din cm

Examples of problem solving

Example

Exercise. Figure 1 shows a body that has an axis of rotation OO". The moment of force applied to the body relative to a given axis will be equal to zero? The axis and the force vector are located in the plane of the figure.

Solution. As a basis for solving the problem, we will take the formula that determines the moment of force:

In the vector product (can be seen from the figure). The angle between the force vector and the radius vector will also be different from zero (or), therefore, the vector product (1.1) is not equal to zero. This means that the moment of force is different from zero.

Answer.

Example

Exercise. The angular velocity of a rotating rigid body changes in accordance with the graph shown in Fig. 2. At which of the points indicated on the graph is the moment of forces applied to the body equal to zero?