The moment of force about the axis or simply the moment of force is the projection of the force onto a straight line that 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 force on 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 force on the body. The axis in this case is the place where the body is attached, relative to which it can rotate. If the body is not fixed, then the center of mass can be considered the axis of rotation.
Formula 1 - Moment of force.
F - The force acting on the body.
r - Shoulder strength.
Figure 1 - Moment of force.
As can be seen from the figure, the shoulder of the force is the distance from the axis to the point of application of the force. But this is the case 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. And moving the point of application of the force along the direction of the force does not change its momentum.
It is customary to consider positive such a moment of force, which causes the body to rotate clockwise relative to the point of observation. 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 the line passing 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 forward along the line of application.
Figure 2 - The moment of force is zero.
If several forces act on the body, then the moment of force will be determined by their resultant. For example, two forces equal in magnitude and directed oppositely 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. In simple terms, 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 1
The moment of force is a torque or rotational moment, while being a vector physical quantity.
It is defined as the vector product of the force vector and also the radius vector, which is drawn from the axis of rotation to the point of application of the specified force.
The moment of force is a characteristic of the rotational effect of force on a rigid body. The concepts of "rotating" and "torque" moments will not be considered identical, since in technology the concept of "rotating" moment is considered as an external force applied to an object.
At the same time, the concept of “torque” is considered in the format of an internal force that occurs in an object under the influence of certain applied loads (a similar concept is used for the resistance of materials).
Concept of moment of force
The moment of force in physics can be considered as the so-called "rotating force". The SI unit of measure is the newton meter. The moment of force can also be called the "moment of a pair of forces", as noted in the works of Archimedes on levers.
Remark 1
In simple examples, when a force is applied to a lever in a perpendicular relation to it, the moment of force will be determined as the product of the magnitude of the specified force and the distance to the axis of rotation of the lever.
For example, a force of three newtons applied at a distance of two meters from the axis of rotation of the lever creates a moment equivalent to a force of one newton applied at a distance of 6 meters to the lever. More precisely, the moment of force of a particle is determined in the format of a vector product:
$\vec (M)=\vec(r)\vec(F)$, where:
- $\vec (F)$ represents the force acting on the particle,
- $\vec (r)$ is the radius of the particle vector.
In physics, energy should be understood as a scalar quantity, while the moment of force will be considered a (pseudo) vector quantity. The coincidence of the dimensions of such quantities will not be accidental: a moment of force of 1 N m, which is applied through a whole revolution, performing mechanical work, imparts an energy of 2 $\pi$ joules. Mathematically it looks like this:
$E = M\theta $, where:
- $E$ represents energy;
- $M$ is considered to be a torque;
- $\theta $ will be the angle in radians.
Today, the measurement of the moment of force is carried out by using special load sensors of strain gauge, optical and inductive types.
Formulas for calculating the moment of force
Interesting in physics is the calculation of the moment of force in the field, produced by the formula:
$\vec(M) = \vec(M_1)\vec(F)$, where:
- $\vec(M_1)$ is considered to be the moment of the lever;
- $\vec(F)$ represents the magnitude of the acting force.
The disadvantage of such a representation is the fact that it does not determine the direction of the moment of force, but only its magnitude. When the force is perpendicular to the vector $\vec(r)$, the moment of the lever will be equal to the distance from the center to the point of the applied force. In this case, the moment of force will be maximum:
$\vec(T)=\vec(r)\vec(F)$
When a force performs a certain action at any distance, it will perform mechanical work. In the same way, the moment of force (when performing an action through an angular distance) will do work.
$P = \vec (M)\omega $
In the existing international measurement system, power $P$ will be measured in Watts, and the moment of force itself will be measured in Newton meters. In this case, the angular velocity is determined in radians per second.
Moment of several forces
Remark 2
When a body is exposed to two equal, as well as oppositely directed forces that do not lie on the same straight line, there is a lack of stay of this body in a state of equilibrium. This is explained by the fact that the resulting moment of these forces relative to any of the axes does not have a zero value, since both presented forces have moments directed in the same direction (pair of forces).
In a situation where the body is fixed on the axis, it will rotate under the influence of a pair of forces. If a pair of forces is applied to a free body, then it will begin to rotate around an axis passing through the center of gravity of the body.
The moment of a pair of forces is considered to be the same with respect to any axis that is perpendicular to the plane of the pair. In this case, the total moment $M$ of the pair will always be equal to the product of one of the forces $F$ and the distance $l$ between the forces (the arm of the pair), regardless of the types of segments into which it divides the position of the axis.
$M=(FL_1+FL-2) = F(L_1+L_2)=FL$
In a situation where the resultant of the moment of several forces is equal to zero, it will be considered the same with respect to all axes parallel to each other. For this reason, the impact on the body of all these forces can be replaced by the action of just one pair of forces with the same moment.
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.
Decision. 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?
Moment of force relative to an arbitrary center in the plane of action of the force, the product of the modulus of force and the arm 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, anti-clockwise-plus;
The moment of force can be expressed as a vector. This is a 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 force systems.
Consider the moment of force about the axis, calculate the moment of force about the Z axis;
Project F onto XY;
F xy =F cosα= ab
m 0 (F xy)=m z (F), i.e. m z =F xy * h= F cosα* h
The moment of force about the axis is equal to the moment of its projection onto a plane perpendicular to the axis, taken at the intersection of the axes and the plane
If the force is parallel to the axis or crosses it, then m z (F)=0
Expression of the moment of force as a vector expression
Draw r a to point A. Consider OA x F.
This is the third vector m o perpendicular to the plane. The cross product modulus can be calculated using twice the area of the shaded triangle.
Analytical expression of force relative to the coordinate axes.
Suppose that the Y and Z, X axes are associated with point O with unit vectors i, j, k Considering that:
r x = X * Fx ; r y = Y * F y ; r z =Z * F y we get: m o (F)=x =
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 moment vector on the axis, and then the moment vector itself.
Varignon's theorem on the moment of the resultant
If the 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 links. For this, the basic equations in a flat system of forces are used.
2 or 3 moment equations can be used.
Example
Let's make an equation for the sum of all forces on the X and Y axis:
The sum of the moments of all forces about point A:
Parallel Forces
Equation for point A:
Equation for point B:
The sum of the projections of forces on the Y axis.
Rotation is a typical type of mechanical movement that is often found in nature and technology. Any rotation arises as a result of the action of some external force on the system under consideration. This force creates the so-called What it is, what it depends on, is discussed in the article.
Rotation process
Before considering the concept of torque, let's characterize the systems to which this concept can be applied. The system of rotation assumes the presence in it of an axis around which a circular movement or rotation is carried out. The distance from this axis to the material points of the system is called the radius of rotation.
From the point of view of kinematics, the process is characterized by three angular quantities:
- rotation angle θ (measured in radians);
- angular velocity ω (measured in radians per second);
- angular acceleration α (measured in radians per square second).
These quantities are related to each other by the following equalities:
Examples of rotation in nature are the movements of planets in their orbits and around their axes, the movements of tornadoes. In everyday life and technology, the movement in question is typical for engine motors, wrenches, construction cranes, opening doors, and so on.
Determining the moment of force
Now let's move on to the actual topic of the article. According to the physical definition, it is the vector product of the vector of force application relative to the axis of rotation and the vector of the force itself. The corresponding mathematical expression can be written as follows:
Here the vector r¯ is directed from the axis of rotation to the point of application of the force F¯.
In this torque formula M¯, the force F¯ can be directed in any direction relative to the direction of the axis. However, the axis-parallel force component will not create rotation if the axis is rigidly fixed. In most problems in physics, one has to consider the forces F¯, which lie in planes perpendicular to the axis of rotation. In these cases, the absolute value of the torque can be determined from the following formula:
|M¯| = |r¯|*|F¯|*sin(β).
Where β is the angle between the vectors r¯ and F¯.
What is a power lever?
The lever of force plays an important role in determining the magnitude of the moment of force. To understand what we are talking about, consider the following figure.
Here is shown a rod of length L, which is fixed at a pivot point by one of its ends. The other end is acted upon by a force F directed at an acute angle φ. According to the definition of the moment of force, we can write:
M \u003d F * L * sin (180 o -φ).
The angle (180 o -φ) appeared because the vector L¯ is directed from the fixed end to the free one. Given the periodicity of the trigonometric sine function, we can rewrite this equality in the following form:
Now let's pay attention to a right triangle built on sides L, d and F. By definition of the sine function, the product of the hypotenuse L and the sine of the angle φ gives the value of the leg d. Then we come to equality:
The linear value d is called the lever of force. It is equal to the distance from the force vector F¯ to the axis of rotation. As can be seen from the formula, it is convenient to use the concept of a force lever when calculating the moment M. The resulting formula says that the maximum torque for some force F will occur only when the length of the radius vector r¯ (L¯ in the figure above) is equal to force lever, that is, r¯ and F¯ will be mutually perpendicular.
Direction of action of the quantity M¯
It was shown above that torque is a vector characteristic for a given system. Where is this vector directed? It is not difficult to answer this question if we remember that the result of the product of two vectors is the third vector, which lies on an axis perpendicular to the plane of the original vectors.
It remains to decide whether the moment of force will be directed upwards or downwards (toward or away from the reader) relative to the said plane. You can determine this either by the gimlet rule, or by using the right hand rule. Here are both rules:
- Right hand rule. If the right hand is positioned in such a way that its four fingers move from the beginning of the vector r¯ to its end, and then from the beginning of the vector F¯ to its end, then the protruding thumb will indicate the direction of the moment M¯.
- The gimlet rule. If the direction of rotation of an imaginary gimlet coincides with the direction of rotational motion of the system, then the translational movement of the gimlet will indicate the direction of the vector M¯. Recall that it only rotates clockwise.
Both rules are equal, so everyone can use the one that is more convenient for him.
When solving practical problems, the different direction of the torque (up - down, left - right) is taken into account using the signs "+" or "-". It should be remembered that the positive direction of the moment M¯ is considered to be the one that leads to the rotation of the system counterclockwise. Accordingly, if some force leads to the rotation of the system in the direction of the clock, then the moment created by it will have a negative value.
The physical meaning of M¯
In the physics and mechanics of rotation, the quantity M¯ determines the ability of a force or the sum of forces to rotate. Since the mathematical definition of the quantity M¯ contains not only force, but also the radius vector of its application, it is the latter that largely determines the noted rotational ability. To make it clearer what ability we are talking about, here are a few examples:
- Every person, at least once in his life, tried to open the door, not by holding the handle, but by pushing it close to the hinges. In the latter case, you have to make a significant effort to achieve the desired result.
- To unscrew the nut from the bolt, use special wrenches. The longer the wrench, the easier it is to unscrew the nut.
- In order to feel the importance of the lever of power, we suggest that readers do the following experiment: take a chair and try to hold it with one hand in weight, in one case, lean the hand against the body, in the other, perform the task on a straight arm. The latter will be an overwhelming task for many, although the weight of the chair has remained the same.
Units of moment of force
A few words should also be said about the units in which torque is measured in SI. According to the formula written for it, it is measured in newtons per meter (N * m). However, these units also measure work and energy in physics (1 N*m = 1 joule). The joule for the moment M¯ does not apply, since work is a scalar quantity, while M¯ is a vector.
Nevertheless, the coincidence of the units of the moment of force with the units of energy is not accidental. The work on the rotation of the system, done by the moment M, is calculated by the formula:
From where we get that M can also be expressed in joules per radian (J/rad).
Rotation dynamics
At the beginning of the article, we wrote down the kinematic characteristics that are used to describe the movement of rotation. In rotational dynamics, the main equation that uses these characteristics is the following:
The action of the moment M on a system having a moment of inertia I leads to the appearance of an angular acceleration α.
This formula is used to determine the angular frequencies of rotation in technology. For example, knowing the torque of an asynchronous motor, which depends on the frequency of the current in the stator coil and on the magnitude of the changing magnetic field, as well as knowing the inertial properties of the rotating rotor, it is possible to determine to what rotation speed ω the motor rotor spins in a known time t.
Problem solution example
A weightless lever, the length of which is 2 meters, has a support in the middle. What weight should be put on one end of the lever so that it is in a state of equilibrium, if on the other side of the support at a distance of 0.5 meters from it lies a mass of 10 kg?
Obviously, what will come if the moments of forces created by the loads are equal in absolute value. The force that creates the moment in this problem is the weight of the body. The levers of force are equal to the distances from the weights to the support. We write the corresponding equality:
m 1 *g*d 1 = m 2 *g*d 2 =>
P 2 \u003d m 2 * g \u003d m 1 * g * d 1 / d 2.
The weight P 2 will be obtained if we substitute from the condition of the problem the values m 1 = 10 kg, d 1 = 0.5 m, d 2 = 1 m. The written equality gives the answer: P 2 = 49.05 newtons.