The moment of power is brief. Statics
In physics, the consideration of problems with rotating bodies or systems that are in equilibrium is carried out using the concept of "moment of force". This article will consider the formula for the moment of force, as well as its use for solving the specified type of problems.
in physics
As noted in the introduction, this article will focus on systems that can rotate either around an axis or around a point. Consider an example of such a model, shown in the figure below.
We see that the gray lever is fixed on the axis of rotation. At the end of the lever there is a black cube of some mass on which a force acts (red arrow). It is intuitively clear that the effect of this force will be the rotation of the lever around the axis counterclockwise.
The moment of force is a quantity in physics, which is equal to the vector product of the radius connecting the axis of rotation and the point of application of the force (green vector in the figure), and the external force itself. That is, the forces about the axis are written as follows:
The result of this product is the vector M¯. Its direction is determined based on the knowledge of the vector-factors, that is, r¯ and F¯. According to the definition of a vector product, M¯ must be perpendicular to the plane formed by the vectors r¯ and F¯, and directed in accordance with the rule of the right hand (if four fingers of the right hand are placed along the first multiplied vector towards the end of the second, then the thumb will indicate where the desired vector is directed). In the figure, you can see where the vector M¯ is directed (blue arrow).
Scalar notation M¯
In the figure in the previous paragraph, the force (red arrow) acts on the lever at an angle of 90 o. In general, however, it can be applied at absolutely any angle. Consider the image below.
Here we see that the force F is already acting on the lever L at a certain angle Φ. For this system, the formula for the moment of force relative to a point (shown by an arrow) in scalar form will take the form:
M = L * F * sin (Φ)
From the expression it follows that the moment of force M will be the greater, the closer the direction of action of force F is to an angle of 90 o with respect to L. On the contrary, if F acts along L, then sin (0) = 0, and the force does not create any moment ( M = 0).
When considering the moment of force in scalar form, the concept of "lever of force" is often used. This value is the distance between the axis (pivot point) and the vector F. Applying this definition to the figure above, we can say that d = L * sin (Φ) is a lever of force (equality follows from the definition of the trigonometric function "sine"). Through the lever of force, the formula for the moment M can be rewritten as follows:
The physical meaning of the quantity M
The considered physical quantity determines the ability of the external force F to exert a rotational effect on the system. To bring the body into rotational motion, it must be imparted some moment M.
A prime example of this process is opening or closing a door to a room. Grasping the handle, the person makes an effort and turns the door on its hinges. Everyone can do it. If you try to open the door by acting on it near the hinges, then you will need to make great efforts to move it from its place.
Another example is loosening a nut with a wrench. The shorter this key is, the more difficult it is to complete the task.
These features are demonstrated by the force over the shoulder, which was given in the previous paragraph. If M is considered constant, then the smaller d, the larger F should be applied to create a given moment of force.
Several acting forces in the system
Above, we considered the cases when only one force F acts on a system capable of rotation, but what to do when there are several such forces? Indeed, this situation is more frequent, since forces of various nature (gravitational, electrical, friction, mechanical, and others) can act on the system. In all these cases, the resulting moment of force M¯ can be obtained using the vector sum of all the moments M i ¯, that is:
M¯ = ∑ i (M i ¯), where i is the number of the force F i
An important conclusion follows from the property of additivity of moments, which is called the Varignon theorem, named after the late 17th - early 18th century mathematician, Frenchman Pierre Varignon. It reads: "The sum of the moments of all the forces affecting the system under consideration can be represented as a moment of one force, which is equal to the sum of all the others and is applied to a certain point." Mathematically, the theorem can be written as follows:
∑ i (M i ¯) = M¯ = d * ∑ i (F i ¯)
This important theorem is often used in practice to solve problems on the rotation and balance of bodies.
Does a moment of power do the work?
Analyzing the above formulas in scalar or vector form, we can come to the conclusion that the value of M is some work. Indeed, its dimension is equal to H * m, which in SI corresponds to joule (J). In fact, a moment of force is not a work, but only a quantity that is capable of performing it. For this to happen, it is necessary to have a circular motion in the system and a long-term action M. Therefore, the formula for the work of the moment of force is written in the following form:
In this expression, θ is the angle at which the rotation was made by the moment of force M. As a result, the unit of work can be written as N * m * rad or J * rad. For example, a value of 60 J * rad indicates that during a rotation of 1 radian (approximately 1/3 of a circle), the force F creating the moment M did a work of 60 joules. This formula is often used when solving problems in systems where frictional forces act, which will be shown below.
Moment of force and moment of impulse
As shown, the impact on the system of the moment M leads to the appearance of rotational motion in it. The latter is characterized by a quantity that is called "angular momentum". It can be calculated using the formula:
Here I is the moment of inertia (a quantity that plays the same role in rotation as mass in the linear motion of a body), ω is the angular velocity, it is related to the linear velocity by the formula ω = v / r.
Both moments (momentum and force) are related to each other by the following expression:
M = I * α, where α = dω / dt - angular acceleration.
Here is another formula that is important for solving problems for the work of moments of forces. Using this formula, you can calculate the kinetic energy of a rotating body. It looks like this:
Balance of several bodies
The first task is related to the equilibrium of a system in which several forces are acting. The figure below shows a system that is acted upon by three forces. It is necessary to calculate how much mass the object needs to be suspended from this lever and at what point it should be done so that this system is in equilibrium.
From the condition of the problem, it can be understood that to solve it, one should use the Varignon theorem. The first part of the problem can be answered immediately, since the weight of the object that should be suspended from the lever will be equal to:
P = F 1 - F 2 + F 3 = 20 - 10 + 25 = 35 N
The signs are chosen here taking into account that the force rotating the lever counterclockwise creates a negative moment.
The position of point d, where this weight should be suspended, is calculated by the formula:
M 1 - M 2 + M 3 = d * P = 7 * 20 - 5 * 10 + 3 * 25 = d * 35 => d = 165/35 = 4.714 m
Note that using the formula for the moment of gravity, we have calculated the equivalent value M of that created by the three forces. For the system to be in equilibrium, it is necessary to suspend a body weighing 35 N at a point 4.714 m from the axis on the other side of the arm.
Moving disc problem
The solution to the next problem is based on the use of the formula for the frictional moment and the kinetic energy of a body of revolution. Problem: given a disc of radius r = 0.3 meters, which rotates at a speed of ω = 1 rad / s. It is necessary to calculate how much distance it can travel on the surface if the rolling friction coefficient is μ = 0.001.
The easiest way to solve this problem is to use the law of conservation of energy. We have the initial kinetic energy of the disk. When it starts rolling, then all this energy is spent on heating the surface due to the action of the friction force. Equating both values, we get the expression:
I * ω 2/2 = μ * N / r * r * θ
The first part of the formula is the kinetic energy of the disk. The second part is the work of the frictional moment F = μ * N / r applied to the edge of the disk (M = F * r).
Considering that N = m * g and I = 1 / 2m * r 2, we calculate θ:
θ = m * r 2 * ω 2 / (4 * μ * m * g) = r 2 * ω 2 / (4 * μ * g) = 0.3 2 * 1 2 / (4 * 0.001 * 9.81 ) = 2.29358 glad
Since 2pi radians correspond to a length of 2pi * r, then we get that the desired distance that the disk will travel is:
s = θ * r = 2.29358 * 0.3 = 0.688 m or about 69 cm
Note that this result is not affected by the mass of the disk.
The rule of the lever, 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.
The rule of the moment
The concept of the moment of forces was introduced. The moment of force is a physical quantity equal to the product of the force on her shoulder:
where M is the moment of force,
F - strength,
l is the shoulder of force.
From the balance rule of the lever directly the rule of moments of forces follows:
F1 / F2 = l2 / l1 or, according to the proportion property F1 * l1 = F2 * l2, that is, M1 = M2
In verbal expression, the rule of moments of forces sounds as follows: the lever is in equilibrium under the action of two forces if the moment of force rotating it clockwise is equal to the moment of force rotating it counterclockwise. The rule of moments of forces is valid for any body fixed around a fixed axis. In practice, the moment of force is found as follows: the line of action of the force is drawn in the direction of the action of the force. 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 shoulder of the force. Multiplying the value of the modulus of force by its shoulder, 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 action of the force depends both on the force itself and on its shoulder.
Application of the rule of moments of forces in different situations
Hence 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. You can do an elementary experiment and make sure that it is easier to push the door, the further we apply force from the axis of rotation. A practical experiment in this case is directly confirmed by the formula. Since, in order for the moments of forces at different shoulders to be equal, it is necessary that the larger shoulder corresponds to a smaller force and vice versa, a larger one corresponds to the smaller shoulder. The closer to the axis of rotation we apply the force, the more it should be. The further from the axis we act with the lever, rotating the body, the less force we will need to apply. Numerical values are easily found from the formula for the rule of moments.
It is on the basis of the rule of moments of forces that we take a crowbar or a long stick, if we need to lift something heavy, and, putting 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.
The moment of force about the axis or simply the moment of force is called the projection of the 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 force on the shoulder of its application. The shoulder in this case is the distance from the axis to the point of application of the force. The moment of force characterizes the rotational action of the force on the body. The axis in this case is the place of attachment of the body, 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 Power.
F - Force acting on the body.
r - Shoulder force.
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 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 to it from the axis. The length of this perpendicular will be equal to the shoulder of the force. And the movement of the point of application of the force along the direction of the force does not change its moment.
It is considered to be a positive moment of force that causes the body to turn 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 a shoulder of 1 meter.
If the force acting on the body passes along a 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. That moment of force in this case will be equal to zero. Since this force will not cause the body to rotate, but will simply move it translationally along the line of application.
Picture 2 - The moment of force is equal to 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 the 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 another with the same force against it, then the carousel will remain motionless.
A moment of power relative to an arbitrary center in the plane of action of the force, is called the product of the modulus of the force on the shoulder.
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.
Sign of the moment:
Clockwise minus, counterclockwise plus;
The moment of force can be expressed as a vector. This is perpendicular to the plane according to the 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.
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), that is, m z = F xy * h= F cosα* h
The moment of force about 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 crosses it, then m z (F) = 0
Expression of the moment of force in the form of 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 modulus of a cross product can be calculated using twice the area of the shaded triangle.
Analytical expression of the force relative to the coordinate axes.
Suppose that the Y and Z, X axes with unit vectors i, j, k are connected to the 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 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 equal to equilibrate.
The sum of the moments relative to 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 one plane.
The purpose of calculating problems of this type is to determine the reactions of external relations. For this, the basic equations in the plane system of forces are used.
2 or 3 equations of moments can be used.
Example
Let's make an equation for the sum of all forces on the X and Y axes:
The sum of the moments of all forces relative to point A:
Parallel forces
Equation relative to point A:
Equation about point B:
The sum of the projections of forces on the Y-axis.
Which is equal to the product of the force on her shoulder.
The moment of force is calculated using the formula:
where F- force, l- the shoulder of strength.
Shoulder of strength 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 drawing and passes through the point, which is designated as the letter O. Shoulder force F t here is the distance l, from the axis of rotation to the line of action of the force. Define it this way. The first step is to draw the line of action of the force, then from the point O, through which the axis of rotation of the body passes, a perpendicular is lowered onto the line of action of the force. The length of this perpendicular turns out to be the arm of the given force.
The moment of force characterizes the rotating action of the force. This action depends on both strength and shoulder. The larger the shoulder, the less force must be applied in order to obtain the desired result, that is, the same moment of force (see the figure above). That is why it is much more difficult to open the door by pushing it near the hinges than by grasping the handle, and it is much easier to unscrew the nut with a long wrench than with a short wrench.
The unit of moment of force in SI is a moment of force of 1 N, the shoulder of which is equal to 1m - Newton-meter (Nm).
The rule of the moments.
A rigid body that can rotate around a 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 one straight line, then such a body is not in equilibrium, since the resulting moment of these forces relative to any axis does not equal zero, since both forces have moments directed in the same direction ... Two such forces acting simultaneously on the body are called with a couple of forces... If the body is fixed on an axis, then it will rotate under the action of a pair of forces. If a pair of forces is applied to a “free body, then it will rotate around an 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. Cumulative moment M pair is always equal to the product of one of the forces F at a distance l between forces called shoulder pair, no matter what segments l, and shares the position of the axis with the shoulder of the pair:
The moment of several forces, the resultant of which is equal to zero, will be the same with respect 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.
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