Statics. Moment of power

When solving problems of moving objects, in some cases their spatial dimensions are neglected, introducing the concept of a material point. For another type of problems, in which bodies at rest or rotating bodies are considered, it is important to know their parameters and the points of application of external forces. In this case, we are talking about the moment of forces about the axis of rotation. Let's consider this issue in the article.

The concept of the moment of force

Before bringing about a fixed axis of rotation, it is necessary to clarify what phenomenon will be discussed. Below is a figure that shows a wrench of length d, a force F is applied to its end. It is easy to imagine that the result of its action will be the rotation of the wrench counterclockwise and unscrewing the nut.

According to the definition, the moment of force is relatively the product of the shoulder (d in this case) and the force (F), that is, the following expression can be written: M = d * F. It should immediately be noted that the above formula is written in scalar form, that is, it allows you to calculate the absolute value of the moment M. As can be seen from the formula, the unit of measurement of the considered quantity is newtons per meter (N * m).

- vector quantity

As discussed above, the moment M is actually a vector. To clarify this statement, consider another figure.

Here we see a lever of length L, which is fixed on the axis (shown by the arrow). A force F is applied to its end at an angle Φ. It is not difficult to imagine that this force will cause the lever to rise. The formula for the moment in vector form in this case will be written as follows: M¯ = L¯*F¯, here the line over the symbol means that the quantity in question is a vector. It should be clarified that L¯ is directed from the axis of rotation to the point of application of the force F¯.

The above expression is a vector product. Its resulting vector (M¯) will be perpendicular to the plane formed by L¯ and F¯. To determine the direction of the moment M¯, there are several rules (right hand, gimlet). In order not to memorize them and not get confused in the order of multiplication of the vectors L¯ and F¯ (the direction of M¯ depends on it), you should remember one simple thing: the moment of force will be directed in such a way that if you look from the end of its vector, then the acting force F ¯ will rotate the lever counterclockwise. This direction of the moment is conditionally taken as positive. If the system rotates clockwise, then the resulting moment of forces has a negative value.

Thus, in the considered case with the lever L, the value of M¯ is directed upwards (from the figure to the reader).

In scalar form, the formula for the moment is written as: M = L*F*sin(180-Φ) or M = L*F*sin(Φ) (sin(180-Φ) = sin(Φ)). According to the definition of the sine, we can write the equality: M = d*F, where d = L*sin(Φ) (see the figure and the corresponding right triangle). The last formula is similar to the one given in the previous paragraph.

The above calculations demonstrate how to work with vector and scalar quantities of moments of forces in order to avoid errors.

The physical meaning of M¯

Since the two cases considered in the previous paragraphs are associated with rotational motion, one can guess what meaning the moment of force carries. If the force acting on a material point is a measure of the increase in the speed of the linear displacement of the latter, then the moment of force is a measure of its rotational ability in relation to the system under consideration.

Let's take an illustrative example. Any person opens the door by holding its handle. It can also be done by pushing the door in the area of ​​the handle. Why doesn't anyone open it by pushing in the hinge area? Very simple: the closer the force is applied to the hinges, the more difficult it is to open the door, and vice versa. The derivation of the previous sentence follows from the formula for the moment (M = d*F), which shows that when M = const, the quantities d and F are inversely related.

Moment of force - additive quantity

In all the cases considered above, there was only one acting force. When solving real problems, the situation is much more complicated. Usually systems that rotate or are in equilibrium are subject to several torsional forces, each of which creates its own moment. In this case, the solution of problems is reduced to finding the total moment of forces relative to the axis of rotation.

The total moment is found by the usual sum of the individual moments for each force, however, remember to use the correct sign for each of them.

Problem solution example

To consolidate the acquired knowledge, it is proposed to solve the following problem: it is necessary to calculate the total moment of force for the system shown in the figure below.

We see that three forces (F1, F2, F3) act on a lever 7 m long, and they have different points of application relative to the axis of rotation. Since the direction of forces is perpendicular to the lever, there is no need to use a vector expression for the moment of torsion. It is possible to calculate the total moment M using a scalar formula and remembering to set the desired sign. Since the forces F1 and F3 tend to turn the lever counterclockwise, and F2 - clockwise, the moment of rotation for the first will be positive, and for the second - negative. We have: M \u003d F1 * 7-F2 * 5 + F3 * 3 \u003d 140-50 + 75 \u003d 165 N * m. That is, the total moment is positive and directed upwards (at the reader).

Definition

The vector product of the radius - vector (), which is drawn from the 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 the point O:

In Fig. 1, the point O and the force vector () and the 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 figure and has a direction away from us. The vector of the moment of force is axial. The direction of the vector of the moment of force is chosen in such a way that the rotation around the point O in the direction of the force and the vector create a right screw system. The direction of the moment of forces and angular acceleration are the same.

The value of the vector is:

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

Moment of force about the axis

The moment of force with respect to the 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 the given axis. In this case, the choice of the point does not matter.

The main moment of forces

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

In this case, the 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 reduction of forces (O and O '), then they are related by the expression:

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

In the general case, the result of the action on a rigid body of an arbitrary system of forces 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).

The basic law of the dynamics of rotational motion

where is the angular momentum of the rotating body.

For a rigid body, this law can be represented as:

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

Units of moment of force

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

To CGS: [M]=dyn 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 about a given axis will be equal to zero? The axis and force vector are located in the plane of the figure.

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

In a vector product (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, which is 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?

Denoting the moment of force relative to the axes , and , we can write:

where , and modules of projections of forces on planes perpendicular to the axis relative to which the moment is determined; l - shoulders equal in length

perpendiculars from the point of intersection of the axis with the plane to the projection or its continuation; the plus or minus sign is placed depending on which direction the shoulder is turning l the projection vector, if you look at the projection plane from the positive direction of the axis; when the projection vector tends to rotate the arm counterclockwise, we agree to consider the moment as positive, and vice versa.

Hence, moment of force about the axis called an algebraic (scalar) quantity equal to the moment of the projection of the force onto a plane perpendicular to the axis, relative to the point of intersection of the axis with the plane.

The previous figure illustrates the sequence of determining the moment of force about the Z-axis. If the force is given and the axis is selected (or specified), then: a) a plane is selected perpendicular to the axis (the XOY plane); b) the force F is projected onto this plane and the module of this projection is determined; c) from the point 0 of the intersection of the axis with the plane, the perpendicular OS to the projection is lowered and the shoulder l = OS is determined; d) looking at the XOU plane from the positive direction of the Z axis (i.e., in this case, from above), we see that the OS is rotated by the vector against the clock, which means

The moment of force about the axis is zero if the force and the axis lie in the same plane: a) the force intersects the axis (in this case l = 0);

b) the force is parallel to the axis ();

c) the force acts along the axis ( l=0 and ).

Spatial system of arbitrarily located forces.

Equilibrium condition

Previously, the process of bringing forces to a point was described in detail and it was proved that any flat system of forces is reduced to a force - the main vector and a pair, the moment of which is called the main moment, and the force and pair equivalent to this system of forces act in the same plane as the given system. This means that if the main moment is represented as a vector, then the main vector and the main moment of a plane system of forces are always perpendicular to each other.

Arguing similarly, one can consistently bring to the point of force of the spatial system. But now the main vector is the closing vector of the spatial (rather than flat) force polygon; the main moment can no longer be obtained by algebraic addition of the moments of these forces with respect to the reduction point. When reduced to a point of a spatial system of forces, the attached pairs act in different planes, and it is advisable to represent their moments in the form of vectors and add them geometrically. Therefore, the main vector (the geometric sum of the forces of the system) and the main moment (the geometric sum of the moments of forces relative to the point of reduction) obtained as a result of the reduction of the spatial system of forces, generally speaking, are not perpendicular to each other.

Vector equalities and express the necessary and sufficient condition for the equilibrium of a spatial system of arbitrarily located forces.

If the main vector is equal to zero, then its projections on three mutually perpendicular axes are also equal to zero. If the main moment is equal to zero, then three of its components on the same axis are equal to zero.

This means that an arbitrary spatial system of forces is statically determinable only if the number of unknowns does not exceed six.

Among the problems of statics, there are often those in which a spatial system of forces parallel to each other acts on the body.

In a spatial system of parallel forces, there should be no more than three unknowns, otherwise the problem becomes statically indeterminate.

Chapter 6

Basic concepts of kinematics

The branch of mechanics that studies the motion of material bodies without taking into account their masses and the forces acting on them is called kinematics.

Movement- the main form of existence of the entire material world, peace and balance- special cases.

Any movement, including mechanical movement, occurs in space and time.

All bodies consist of material points. To get a correct idea of ​​the motion of bodies, you need to start studying with the motion of a point. The movement of a point in space is expressed in meters, as well as in submultiple (cm, mm) or multiples (km) units of length, time - in seconds. In practice or life situations, time is often expressed in minutes or hours. When considering one or another movement of a point, the time is counted from a certain, predetermined initial moment ( t= 0).

The locus of positions of a moving point in the reference frame under consideration is called trajectory. According to the type of trajectory, the movement of a point is divided into rectilinear And curvilinear. The trajectory of a point can be defined and pre-set. So, for example, the trajectories of artificial Earth satellites and interplanetary stations are calculated in advance, or if we take buses moving around the city as material points, then their trajectories (routes) are also known. In such cases, the position of a point at each moment of time is determined by the distance (arc coordinate) S, i.e. the length of the section of the trajectory, counted from some of its fixed points, taken as the origin. The counting of distances from the beginning of the trajectory can be carried out in both directions, therefore, counting in one direction is conditionally taken as positive, and in

opposite - for negative , those. the distance S is an algebraic quantity. It can be positive (S > 0) or negative (S<0).

When moving, a point for a certain period of time passes some path L , which is measured along the path in the direction of travel.

If the point began to move not from the origin O, but from a position at the initial distance S o then

The vector quantity characterizing at any given moment in time the direction and speed of movement of a point is called speed.

The speed of a point at any moment of its movement is directed tangentially to the trajectory.

Note that this vector equality characterizes only the position , and the module of the average velocity over time :

where is the path traveled by the point in time .

The modulus of the average speed is equal to the distance traveled divided by the time during which this path was traveled.

The vector quantity characterizing the speed of change of direction and the numerical value of the speed is called acceleration.

With uniform motion along a curvilinear trajectory, the point also has acceleration, since in this case the direction of speed also changes.

The unit of acceleration is usually taken as .

6.2. Methods for specifying the movement of a point

There are three ways: natural, coordinate, vector.

The natural way to specify the movement of a point. If, in addition to the trajectory on which the origin O is marked, the dependence

between distance S and time t, this equation is called the law of motion of a point along a given trajectory.

Let, for example, some trajectory be given, the movement of a point along which is determined by the equation . Then at time , i.e. the point is at the origin O; at a point in time, the point is at a distance ; at a point in time, the point is at a distance from the origin O.

Coordinate method of specifying point movement. When the trajectory of a point is not known in advance, the position of the point in space is determined by three coordinates: the abscissa X, the ordinate Y, and the applicate Z.

Or excluding time.

These equations express law of motion of a point in a rectangular coordinate system (OXYZ).

In the particular case, if the point moves in a plane, the law of point motion is expressed by two equations: or .

For example. The movement of a point in a plane coordinate system is given by the equations and ( X And Y– cm, t – c). Then at time and , i.e. the point is at the origin; at the point in time the coordinates of the point , ; at the point in time the coordinates of the point , etc.

Knowing the law of motion of a point in a rectangular coordinate system, one can determine point trajectory equation.

For example, by eliminating the time t from the above equations and , we obtain the trajectory equation . As you can see, in this case the point moves along a straight line passing through the origin.

6.3. Determining the speed of a point in a natural way
tasks of her movement

Let point A move along a given trajectory according to the equation , it is required to determine the speed of the point at time t.

For a period of time, the point has traveled a path , the value of the average speed along this path is called tangent, or tangential acceleration. Tangential acceleration modulus

,

equal to the derivative of the speed at a given moment in time, or, otherwise, the second derivative of the distance in time, characterizes the speed of change in the value of the speed.

It is proved that the vector is perpendicular to the tangent at any time, so it is called normal acceleration.

This means that the modulus of normal acceleration is proportional to the second power of the modulus of speed at a given moment, inversely proportional to the radius of curvature of the trajectory at a given point, and characterizes the rate of change in the direction of speed.

Acceleration module

Moment of force about the axis is the moment of the projection of a force onto a plane perpendicular to the axis, relative to the point of intersection of the axis with this plane

The moment about an axis is positive if the force tends to rotate a plane perpendicular to the axis counterclockwise when viewed towards the axis.

The moment of force about the axis is 0 in two cases:

If the force is parallel to the axis

If the force crosses the axis

If the line of action and the axis lie in the same plane, then the moment of force about the axis is 0.

27. The relationship between the moment of force about an axis and the vector moment of force about a point.

Mz(F)=Mo(F)*cosαThe moment of force, relative to the axis, is equal to the projection of the vector of the moment of forces, relative to the point of the axis, on this axis.

28. The main theorem of statics about bringing the system of forces to a given center (Poinsot's theorem). Principal vector and principal moment of the system of forces.

Any spatial system of forces in the general case can be replaced by an equivalent system consisting of one force applied at some point of the body (center of reduction) and equal to the main vector of this system of forces, and one pair of forces, the moment of which is equal to the main moment of all forces relative to the selected referral center.

The main vector of the force system called vector R equal to the vector sum of these forces:

R = F 1 + F 2 + ... + F n= F i .

For a flat system of forces, its main vector lies in the plane of action of these forces.

The main moment of the system of forces about the center O is called the vector L O , equal to the sum of the vector moments of these forces relative to the point O:

L O= M O( F 1) + M O( F 2) + ... + M O( F n) = M O( F i).

Vector R does not depend on the choice of the center O, and the vector L O when changing the position of the center O can generally change.

Poinsot's theorem: An arbitrary spatial system of forces can be replaced by one force with the main vector of the system of forces and a pair of forces with the main moment without disturbing the state of the rigid body. The main vector is the geometric sum of all forces acting on a rigid body and is located in the plane of action of the forces. The main vector is considered through its projections on the coordinate axes.

To bring forces to a given center applied at some point of a rigid body, it is necessary: ​​1) to transfer the force to itself in parallel to a given center without changing the force modulus; 2) in a given center, apply a pair of forces, the vector moment of which is equal to the vector moment of the transferred force of the relative new center, this pair is called an attached pair.

Dependence of the main moment on the choice of the center of reduction. The principal moment relative to the new reduction center is equal to the geometric sum of the principal moment relative to the old reduction center and the cross product of the radius-vector connecting the new reduction center with the old one and the principal vector.

29 Special cases of reducing the spatial system of forces

	Values ​​of principal vector and principal moment	Cast result
		The system of forces is reduced to a pair of forces, the moment of which is equal to the main moment (the main moment of the system of forces does not depend on the choice of the center of reduction O).
		The system of forces is reduced to a resultant equal to passing through the center O.
		The system of forces is reduced to a resultant equal to the main vector and parallel to it and separated from it at a distance. The position of the line of action of the resultant must be such that the direction of its moment relative to the center of reduction O coincides with the direction relative to the center O.
	, and the vectors are not perpendicular	The system of forces is reduced to a dynamo (power screw) - a combination of a force and a pair of forces lying in a plane perpendicular to this force.
		The system of forces applied to a rigid body is balanced.

30. Reduction to dynamism. In mechanics, a dynamo is such a set of forces and a pair of forces () acting on a rigid body, in which the force is perpendicular to the plane of action of the pair of forces. Using the vector moment of a couple of forces, one can also define a dynamo as a combination of a force and a couple whose force is parallel to the vector moment of a couple of forces.

Central helical axis equation Suppose that in the center of reduction, taken as the origin of coordinates, the main vector with projections on the coordinate axes and the main moment with projections are obtained. When the system of forces is reduced to the center of reduction O 1 (Fig. 30), a dynamo is obtained with the main vector and the main moment , Vectors and as forming a linam. are parallel and therefore can differ only by a scalar factor k 0. We have, since .The principal moments and , satisfy the relation

Substituting , we get

The coordinates of the point O 1 in which the dynamo is obtained, we denote x, y, z. Then the projections of the vector on the coordinate axes are equal to the coordinates x, y, z. Given this, (*) can be expressed in the form

where i. j ,k are the unit vectors of the coordinate axes, and the vector product * is represented by the determinant. The vector equation (**) is equivalent to three scalar equations, which, after discarding, can be represented as

The resulting linear equations for the coordinates x, y, z are the equations of a straight line - the central helical axis. Consequently, there is a straight line at the points of which the system of forces is reduced to a dynamo.

In the article we will talk about the moment of force about a point and an axis, definitions, drawings and graphs, what unit of measurement of the moment of force, work and force in rotational motion, as well as examples and tasks.

Moment of power is a vector of physical quantity equal to the product of vectors shoulder strength(radius-vector of the particle) and strength acting on a point. The force lever is a vector connecting the point through which the axis of rotation of the rigid body passes with the point to which the force is applied.

where: r is the shoulder of the force, F is the force applied to the body.

vector direction moment force always perpendicular to the plane defined by the vectors r and F.

Main point- any system of forces on the plane with respect to the accepted pole is called the algebraic moment of the moment of all forces of this system with respect to this pole.

In rotational movements, not only the physical quantities themselves are important, but also how they are located relative to the axis of rotation, that is, their moments. We already know that in rotational motion, not only mass is important, but also. In the case of a force, its effectiveness in triggering acceleration is determined by the way that force is applied to the axis of rotation.

The relationship between power and the way it is used describes MOMENT OF POWER. The moment of force is the vector product of the force arm R to the force vector F:

As in every vector product, so here

Therefore, the force will not affect the rotation when the angle between the force vectors F and lever R is 0 o or 180 o . What is the effect of applying a moment of force M?

We use Newton's second law of motion and the relation between rope and angular velocity v = Rω in scalar form are valid when the vectors R And ω perpendicular to each other

Multiplying both sides of the equation by R, we get

Since mR 2 = I, we conclude that

The above dependence is also valid for the case of a material body. Note that while an external force gives a linear acceleration a, the moment of the external force gives the angular acceleration ε.

Unit of moment of force

The main measure of the moment of force in the SI system coordinate is: [M]=N m

To CGS: [M]=dyn cm

Work and force in rotational motion

Work in linear motion is defined by the general expression,

but in rotation

and consequently

Based on the properties of the mixed product of three vectors, we can write

Therefore, we have an expression for rotary work:

Rotary power:

Find moment of power, acting on the body in the situations shown in the figures below. Assume that r = 1m and F = 2N.

A) since the angle between the vectors r and F is 90°, then sin(a)=1:

M = r F = 1m 2N = 2Nm

b) because the angle between vectors r and F is 0°, so sin(a)=0:

M=0
yes directional force can't give a point rotary motion.

c) since the angle between the vectors r and F is 30°, then sin(a)=0.5:

M = 0.5 r F = 1N m.

Thus, a directional force will cause body rotation, but its effect will be less than in the case a).

Moment of force about the axis

Assume the data is a point O(pole) and power P. At the point O we take the origin of the rectangular coordinate system. Moment of power R in relation to the poles O is a vector M out (R), (picture below) .

Any point A on line P has coordinates (xo, yo, zo).
Force vector P has coordinates Px, Py, Pz. Combining point A (xo, yo, zo) with the beginning of the system, we get a vector p. Force vector coordinates P relative to the pole O marked with symbols Mx, My, Mz. These coordinates can be computed as the minima of the given determinant, where ( i, j, k) are unit vectors on the coordinate axes (options): i, j, k

After solving the determinant, the coordinates of the moment will be equal to:

Moment vector coordinates Mo (P) are called moments of force about the corresponding axis. For example, moment of force P about the axis Oz surrounds the template:

Mz = Pyxo - Pxyo

This pattern is interpreted geometrically as shown in the figure below.

Based on this interpretation, the moment of force about the axis Oz can be defined as the moment of force projection P perpendicular to the axis Oz relative to the point of penetration of this plane by the axis. Force projection P on the perpendicular axis is indicated Pxy , and the penetration point of the plane Oxy- axis OS symbol Oh
From the above definition of the moment of force about an axis, it follows that the moment of force about an axis is zero when the force and the axis are equal, in the same plane (when the force is parallel to the axis, or when the force crosses the axis).
Using formulas on Mx, My, Mz, we can calculate the value of the moment of force P relative to the point O and determine the angles contained between the vector M and system axes:

If the power lies in planes, That zo = 0 and pz = 0 (see picture below).

Moment of power P in relation to the point (pole) O is:
Mx=0,
My = 0
Mo (P) \u003d Mz \u003d Pyxo - Pxy.

Torque mark:
plus (+) - rotation of the force around the O axis clockwise,
minus (-) - rotation of the force around the O axis counterclockwise.