When solving problems of moving objects, in a number of cases their spatial dimensions are neglected, introducing the concept material point. For another type of problem, in which bodies at rest or rotating are considered, it is important to know their parameters and application points external forces. In this case we're talking about about the moment of forces relative to the axis of rotation. Let's look at this issue in the article.
The concept of moment of force
Before bringing it relative to a fixed axis of rotation, it is necessary to explain what phenomenon we'll talk. Below is a drawing 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 influence will be to rotate the wrench counterclockwise and unscrew the nut.
According to the definition, the moment of force about the axis of rotation is the product of the arm (d in in this case) by force (F), that is, we can write the following expression: M = d*F. It should be noted right away that the above formula is written in scalar form, that is, it allows you to calculate absolute value moment M. As can be seen from the formula, the unit of measurement of the value under consideration is newtons per meter (N*m).
- vector quantity
As stated 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 to an 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. Formula for moment in vector form in this case it will be written like this: M¯ = L¯*F¯, here the bar over the symbol means that the quantity in question is a vector. It should be clarified that L¯ is directed from to the point of application of force F¯.
The given expression is a cross product. Its resulting vector (M¯) will be directed 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 to get confused in the order of multiplication of 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 when viewed from the end of its vector, the acting force F¯ will rotate the lever counterclockwise. This direction of the moment is conventionally taken as positive. If the system rotates clockwise, then the resulting moment of force has a negative value.
Thus, in the case under consideration with lever L, the value of M¯ is directed upward (from the figure to the reader).
In scalar form, the formula for the moment will be written as: M = L*F*sin(180-Φ) or M = L*F*sin(Φ) (sin(180-Φ) = sin(Φ)). According to the definition of sine, we can write the equality: M = d*F, where d = L*sin(Φ) (see figure and corresponding right triangle). The last formula is similar to that given in the previous paragraph.
The calculations above demonstrate how to work with vector and scalar quantities moments of strength to prevent mistakes.
Physical meaning of the quantity M¯
Since the two discussed in previous paragraphs cases related to rotational movement, then you can guess what the meaning of the moment of force is. If the force acting on a material point is a measure of the increase in speed linear movement the latter, then the moment of force is a measure of its rotational ability in relation to the system under consideration.
Let's give clear example. Any person opens the door by holding its handle. This can also be done by pushing the door in the handle area. Why doesn't anyone open it by pushing it in the hinge area? It’s very simple: the closer the force is applied to the hinges, the more difficult it is to open the door, and vice versa. The conclusion of the previous sentence follows from the formula for the moment (M = d*F), which shows that at M = const the values of d and F are in inverse relationship.
Moment of force - additive quantity
In all the cases discussed above, there was only one active force. When deciding real problems the situation is much more complicated. Typically, systems that rotate or are in equilibrium are subject to several torsional forces, each of which creates its own torque. In this case, solving 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.
Example of problem solution
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 applications relative to the axis of rotation. Since the direction of the forces is perpendicular to the lever, there is no need to apply vector expression for the torsional moment. You can calculate the total moment M using the scalar formula and not forgetting the formulation the desired sign. Since forces F1 and F3 tend to rotate the lever counterclockwise, and F2 - clockwise, the torque for the first will be positive, and for the second - negative. We have: M = F1*7-F2*5+F3*3 = 140-50+75 = 165 N*m. That is, the total moment is positive and directed upward (toward the reader).
Moment of power (synonyms: torque, torque, torque, torque) - vector physical quantity equal to the vector product of the radius vector drawn from the axis of rotation to the point of application of the force by the vector of this force. Characterizes the rotational action of a force on a solid body.
The concepts of “rotating” and “torque” moments in general case are not identical, since in technology the concept of “rotating” moment is considered as an external force applied to an object, and “torque” is internal force, arising in an object under the influence of applied loads (this concept is used in the resistance of materials).
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7th grade - 39. Moment of force. Rule of Moments
Moment of gravity.Dumbbell and hand
Strength and mass
Moment of power. Levers in nature, technology, everyday life | Physics 7th grade #44 | Info lesson
Dependence of angular acceleration on torque 1
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General information
Special cases
Lever torque formula
Very interesting a special case, represented as a definition of the moment of force in the field:
| M → | = | M → 1 | | F → | (\displaystyle \left|(\vec (M))\right|=\left|(\vec (M))_(1)\right|\left|(\vec (F))\right|), Where: | M → 1 | (\displaystyle \left|(\vec (M))_(1)\right|)- lever moment, | F → | (\displaystyle \left|(\vec (F))\right|)- the magnitude of the acting force.The problem with this representation is that it does not give the direction of the moment of force, but only its magnitude. If the force is perpendicular to the vector r → (\displaystyle (\vec (r))), the lever moment will be equal to the distance to the center and the moment of force will be maximum:
| T → | = | r → | | F → | (\displaystyle \left|(\vec (T))\right|=\left|(\vec (r))\right|\left|(\vec (F))\right|)Force at an angle
If strength F → (\displaystyle (\vec (F))) directed at an angle θ (\displaystyle \theta ) to lever r, then M = r F sin θ (\displaystyle M=rF\sin \theta ).
Static balance
In order for an object to be in equilibrium, not only the sum of all forces must be zero, but also the sum of all moments of force around any point. For a two-dimensional case with horizontal and vertical forces: the sum of forces in two dimensions ΣH=0, ΣV=0 and the moment of force in the third dimension ΣM=0.
Moment of force as a function of time
M → = d L → d t (\displaystyle (\vec (M))=(\frac (d(\vec (L)))(dt))),
Where L → (\displaystyle (\vec (L)))- moment of impulse.
Let's take a solid body. Movement solid can be represented as the movement of a specific point and rotation around it.
The angular momentum relative to point O of a rigid body can be described through the product of the moment of inertia and angular velocity relative to the center of mass and linear motion center of mass
L o → = I c ω → + [ M (r o → − r c →) , v c → ] (\displaystyle (\vec (L_(o)))=I_(c)\,(\vec (\omega )) +)We will consider rotating movements in the Koenig coordinate system, since it is much more difficult to describe the motion of a rigid body in the world coordinate system.
Let's differentiate this expression with respect to time. And if I (\displaystyle I) - constant in time, then
M → = I d ω → d t = I α → (\displaystyle (\vec (M))=I(\frac (d(\vec (\omega )))(dt))=I(\vec (\alpha ))),Where α → (\displaystyle (\vec (\alpha )))- angular acceleration, measured in radians per second per second (rad/s 2). Example: a homogeneous disk rotates.
If the inertia tensor changes with time, then the motion relative to the center of mass is described using the Euler dynamic equation:
M c → = I c d ω → d t + [ w → , I c w → ] (\displaystyle (\vec (M_(c)))=I_(c)(\frac (d(\vec (\omega ))) (dt))+[(\vec (w)),I_(c)(\vec (w))]).
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 attachment point of the body, 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.
Lecture 3. Law of conservation of angular momentum.
Moment of power. Momentum of a material point and mechanical system. Equation of moments of a mechanical system. Law of conservation of angular momentum of a mechanical system.
Mathematical information.
Vector artwork two (non-zero) vectors and is called a vector which in Cartesian system coordinates (with unit vectors , , ) is determined by the formula
.
Value (area of the rectangle on the vectors and ).
Properties vector product.
1) The vector is directed perpendicular to the plane of vectors and. Therefore, for any vector lying in the plane of (linearly independent) vectors and (i.e.), we obtain . Therefore, if two non-zero vectors and parallel, That .
2) The time derivative of the vector product is a vector 
.
Indeed, (basis vectors , , are constant)
Momentum vector
Moment vector momentum relative to point O is called a vector
where is the radius vector from point O, is the momentum vector of the point. The vector is directed perpendicular to the plane of vectors and . Point O is sometimes called pole. Let's find the derivative of the angular momentum vector with respect to time
.
The first term on the right side: 
. Since in inertial system reference according to Newton's second law (in pulse form) , then the second term has the form .
Magnitude 
called a vector moment of force relative to point O.
Finally we get :
the derivative of the angular momentum vector relative to a point is equal to the moment active forces relative to this point.
Properties of the moment of force vector.
.
3) Moment of the sum of forces equal to the sum moments of each force 
.
4) Sum of moments of forces relative to a point
when moving to another point O 1, at which it will change according to the rule
.
Therefore, the moment of force will not change if .
5) Let , where , then 
.
Therefore, if two the same strength lies on one straight line, then their moments the same. This line is called line of action of force. The length of the vector is called the arm of the force relative to points ABOUT.
Moment of force about the axis.
As follows from the definition of moment of force, the coordinates of the vector moments of force relative to coordinate axes are determined by the formulas
, 
, 
.
Let's consider a method for finding the moment of force relative to some z axis To do this, we need to consider the vector of the moment of force relative to a certain point O on this axis and find the projection of the force moment vector onto this axis.
1) The projection of the force moment vector onto the z axis does not depend on the choice of point O.
Let's take two different points O 1 and O 2 on the z axis and find the moments of force F relative to these points.
Vector difference 
is directed perpendicular to the vector lying on the z axis. Therefore, if we consider the unit vector of the z axis – vector, then the projections onto the z axis are equal to each other
Therefore, the moment of force relative to the z axis is uniquely determined.
Consequence. If the moment of force about a certain point on an axis is equal to zero, then the moment of force about this axis is equal to zero.
2) If the force vector is parallel to the z axis, then the moment of force relative to the axis is zero.
Indeed, the vector of the moment of force relative to any point on the axis must be perpendicular to the force vector, therefore it is also perpendicular to the axis parallel to this vector. Therefore, the projection of the force moment vector onto this axis will be equal to zero. Therefore, if the decomposition of the force vector into components parallel to the axis, and component , perpendicular to the axis, That
3) If the force vector and the axis are not parallel, but lie in the same plane, then the moment of force relative to the axis is zero. Indeed, in this case, the vector of the moment of force relative to any point on the axis is directed perpendicular to this plane (since the vector also lies in this plane). You can say it another way. If we consider the point of intersection of the line of action of the force and the straight line z, then the moment of the force about this point is equal to zero, therefore the moment of the force about the axis is equal to zero.
So, to find the moment of force about the z axis, you need to:
1) find the projection of force on any plane p perpendicular to this axis and indicate point O - the point of intersection of this plane with the z axis;
Related information.
The rule of leverage has existed for almost two thousand years, discovered by Archimedes as early as the third century BC, until the seventeenth century from light hand the French scientist Varignon did not receive a more general form.
Torque rule
The concept of torque was introduced. The moment of force is physical quantity, equal to the product strength on her shoulder:
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 sounds in the following way: 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 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 different 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. Can be done elementary experience 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. Numeric values are 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.