Which is equal to the product of the force by its shoulder.
The moment of force is calculated using the formula:
Where F- force, l- shoulder of strength.
Shoulder of power- this is the shortest distance from the line of action of the force to the axis of rotation of the body. The figure below shows a rigid body that can rotate around an axis. The axis of rotation of this body is perpendicular to the plane of the figure and passes through the point, which is designated as the letter O. The shoulder of force Ft here is the distance l, from the axis of rotation to the line of action of the force. It is defined this way. The first step is to draw a line of action of the force, then from point O, through which the axis of rotation of the body passes, lower a perpendicular to the line of action of the force. The length of this perpendicular turns out to be the arm of a given force.
The moment of force characterizes the rotating action of a force. This action is dependent on both strength and leverage. The larger the leverage, the less force must be applied to obtain desired result, that is, the same moment of force (see figure above). That is why it is much more difficult to open a door by pushing it near the hinges than by grasping the handle, and it is much easier to unscrew a nut with a long than with a short wrench.
The SI unit of moment of force is taken to be a moment of force of 1 N, the arm of which is equal to 1 m - newton meter (N m).
Rule of moments.
A rigid body that can rotate around fixed axis, is in equilibrium if the moment of force M 1 rotating it clockwise is equal to the moment of force M 2 , which rotates it counterclockwise:
The rule of moments is a consequence of one of the theorems of mechanics, which was formulated by the French scientist P. Varignon in 1687.
A couple of forces.
If a body is acted upon by 2 equal and oppositely directed forces that do not lie on the same straight line, then such a body is not in equilibrium, since the resulting moment of these forces relative to any axis is not equal to zero, since both forces have moments directed in the same direction . Two such forces simultaneously acting on a body are called a couple of forces. If the body is fixed on an axis, then under the action of a pair of forces it will rotate. If a couple of forces are applied to a free body, then it will rotate around its axis. passing through the center of gravity of the body, figure b.
The moment of a pair of forces is the same about any axis perpendicular to the plane of the pair. Total moment M couples always equal to the product one of the forces F to a distance l between forces, which is called couple's shoulder, no matter what segments l, and shares the position of the axis of the shoulder of the pair:
The moment of several forces, the resultant of which is zero, will be the same relative to all axes parallel to each other, therefore the action of all these forces on the body can be replaced by the action of one pair of forces with the same moment.
Determination of the moment of force relative to a point and an axis. Determination of the force arm relative to a point. Formulations and proofs of the properties of the moment of force. Expression of the absolute value of the moment in the form of the product of the force arm and the force modulus.
ContentThe moment about point O, from a force whose line of action passes through this point, is equal to zero.
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The same applies to forces whose continuation lines intersect at one point. In this case, the point of intersection of the lines of their action is taken as the point of application of the sum of forces.
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The moment of force is pseudovector or, what is the same, axial vector.
This property follows from the property of the vector product. Since the vectors are true(or polar) vectors, then their vector product is pseudovector. This means that we can only determine the absolute value and the axis along which the vector product is directed. We set the direction along this axis arbitrarily, using the rule of the right screw. That is, we mentally plot vectors from the same center. Then turn the handle from position to position. As a result, the right screw moves in a direction perpendicular to the plane in which the vectors are located. We take this direction as the direction of the vector product.
But if we determined the direction using the left screw rule, then the vector product would be directed in the opposite direction. In this case, no contradiction arises. That is, in fact, axial vectors can have two mutually opposite directions. In order not to complicate the mathematical formulas, we select one of them using the right screw rule. For this reason, pseudovectors cannot be geometrically added to true vectors. But they can be multiplied using a scalar or vector product.
Moment of force about the axis
Definition
There are often cases when we do not need to know all the components of the moment of a force about a selected point, but only need to know the moment of a force about a selected axis.
The moment of force about an axis is the projection of the vector of the moment of force about an arbitrary point belonging to this axis onto the direction of the axis.
Let be a unit vector directed along the axis. And let O be an arbitrary point belonging to it. Then the moment of force about the axis is a scalar product:
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This definition is possible because for any two points O and O′ belonging to the axis, the projections of the moments about these points onto the axis are equal. Let's show it.
Let's use the vector equation:
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Let's multiply this equation scalarly by a unit vector directed along the axis:
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Since the vector is parallel to the axis, then . From here
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That is, the projections of moments onto the axis relative to points O and O′ belonging to this axis are equal.
Properties
The moment about an axis due to a force whose line of action passes through this axis is equal to zero.
Proof of properties
Moving the point of application of force along the line of its action
If the point of application of force is moved along the line of action of the force, then the moment, with such movement, will not change.
Proof
Let the force be applied at point A. Through point A we draw a straight line parallel to the force vector. This straight line is the line of its action. Let us move the force application point A to the point A′ belonging to the line of action. Then
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The vector is drawn through two points of the action line. Therefore, its direction coincides or is opposite to the direction of the force vector. Then , where λ is a parameter; . , if point A′ is shifted relative to A in the direction of the vector . Otherwise .
Thus, the vector drawn from O to A′ has the form:
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Let's find the moment of force applied at point A′ using the properties of the vector product:
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We see that the moment has not changed:
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The property has been proven.
Absolute value of the moment of force
Absolute value moment of force relative to a certain point is equal to the product of the absolute value of the force and the shoulder of this force relative to the selected point.
Proof
The absolute value of the moment M relative to point O is equal to the product of force F and its arm d = |OD| .
Let us have a force applied at point A. Let's consider the moment of this force relative to some point O. Note that points O, A and the vector lie in the same plane. Let's depict it in the picture. Through point A, in the direction of the vector, draw a straight line AB. This straight line is called the line of action of the force. Through point O we lower the perpendicular OD to the line of action. And let D be the point of intersection of the line of action and the perpendicular. Then is the force arm relative to the center O. Let's denote it with the letter . Let us use , according to which the point of application of force can be moved along its line of action. Let's move it to point D. Moment of power:
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Since the vectors and are perpendicular, then, by the property of the vector product, the absolute value of the moment:
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where is the absolute value of the force.
Note that the moment vector is perpendicular to the plane of the drawing. Its direction is determined by the right screw rule. If we rotate a screw passing through point O perpendicular to the plane of the figure in the direction of force F, then it will move towards us. Therefore, the moment vector is perpendicular to the plane of the drawing and directed towards us.
The property has been proven.
Moment about a point due to a force passing through that point
The moment about point O, from a force whose line of action passes through this point, is equal to zero.
Proof
Let the line of action of the force pass through point O. Then the shoulder of this force relative to O is equal to zero: . According to, the absolute value of the moment of force relative to the selected point is zero:
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The property has been proven.
Moment of the sum of forces applied at one point
The moment from the vector sum of forces applied to one point of the body is equal to the vector sum of moments from each of the forces applied to the same point:
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Proof
Let the forces be applied at one point A. Let be the vector sum of these forces. We find the moment about some point O from the vector sum applied at point A. To do this, we apply the properties of a vector product:
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The property has been proven.
Moment of a system of forces whose vector sum is zero
If the vector sum of forces is zero:
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then the sum of the moments from these forces does not depend on the position of the center relative to which the moments are calculated:
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Proof
Let the forces be applied at points, respectively. And let the points O and C denote the two centers about which we will calculate the moments. Then the following vector equations hold:
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We use them when calculating the sum of moments about point O:
it is indicated that the moment of force about an axis is the projection of the vector of the moment of force relative to an arbitrary point belonging to this axis onto the direction of the axis. As such a point, we take the point of intersection of the line of action of the force with the axis. But, according to , the moment about this point is equal to zero. Therefore, its projection onto this axis is also zero.
The property has been proven.
Moment about an axis due to a force parallel to this axis
The moment about an axis due to a force parallel to this axis is equal to zero.
Proof
Let O be an arbitrary point on the axis. Let's consider the moment of force about this point. According to the definition:
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According to the cross product property, the moment vector is perpendicular to the force vector. Since the force vector is parallel to the axis, the moment vector is perpendicular to it. Therefore, the projection of the moment about point O onto the axis is zero.
The property has been proven.
Moment of force about the axis is the moment of projection of a force onto a plane perpendicular to an axis, relative to the point of intersection of the axis with this plane
A moment about an axis is positive if the force tends to rotate the plane perpendicular to the axis counterclockwise when looking 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 equal to 0.
27. 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 force, relative to the point of the axis onto this axis.
28. The main theorem of statics about bringing a system of forces to a given center (Poinsot’s theorem). The main vector and the main moment of the system of forces.
In the general case, any spatial system of forces 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 adduction center.
The main vector of the force system called a vector R, equal to the vector sum of these forces:
R = F 1 + F 2 + ... + F n= F i.
For a plane system of forces, its main vector lies in the plane of action of these forces.
The main point of the system of forces relative to the center O is called a vector L O, equal to the sum of the vector moments of these forces relative to 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 center O, and the vector L When the position of the center changes, 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 force system and a pair of forces with a main moment without disturbing the state of the rigid body. The main vector is the geometric sum of all forces acting on a solid 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 solid body, it is necessary: 1) transfer the force parallel to itself to a given center without changing the modulus of the force; 2) at a given center, apply a pair of forces, the vector moment of which is equal to the vector moment of the transferred force relative to the new center; this pair is called an attached pair.
Dependence of the main moment on the choice of the center of reduction. The main moment about the new center of reduction is equal to the geometric sum of the main moment about the old center of reduction and the vector product of the radius vector connecting the new center of reduction with the old one by the main vector.
29 Special cases of reduction of a spatial system of forces
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Principal vector and principal moment values |
Result of casting |
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Force system is reduced to a pair of forces whose moment is equal to the main moment (the main moment of the force system does not depend on the choice of the center of reduction O). |
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The system of forces is reduced to a resultant equal to passing through the center O. |
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The system of forces is reduced to a resultant equal to the main vector and parallel to it and located at a distance from it. 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. |
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The system of forces is reduced to a dyna (power screw) - a combination of force and a pair of forces lying in a plane perpendicular to this force. |
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The system of forces applied to a solid body is balanced. |
, and the vectors are not perpendicular
30. Reduction to dynamism. In mechanics, dynamics is called such a set of forces and pairs of forces () acting on a solid body, in which the force is perpendicular to the plane of action of the pair of forces. Using the vector moment of a pair of forces, we can also define dynamism as the combination of a force and a couple whose force is parallel to the vector moment of the pair of forces.
Equation of the central helical axis Let us assume that at the reduction center, taken as the origin of coordinates, the main vector with projections on the coordinate axes and main point with projections When the system of forces is brought to the center of reduction O 1 (Fig. 30), a dyna is obtained with the main vector and the main moment, Vectors and as forming a linama. are parallel and therefore can differ only in the scalar factor k 0. We have, since the main moments and satisfy the relation
Substituting , we get
Let us denote the coordinates of point O 1 at which the dynamics are obtained as 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 unit vectors of the coordinate axes, and the vector product *is represented by the determinant. The vector equation (**) is equivalent to three scalar ones, 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 dynamism.
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. Direction of the moment of forces and angular acceleration match up.
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 the axis is physical quantity, equal to the projection of the vector of the moment of force relative to the point of the selected axis onto this 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 equal to the sum moments of all forces acting in the system relative 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 solid 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. 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?
