The rule of leverage, 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.
Torque rule
The concept of torque was introduced. The moment of force is a physical quantity equal to the product of the force and its arm:
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 is as follows: 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 of the 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 various 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 a basic experiment 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. Numerical values can be 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.
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. The direction of the moment of forces and angular acceleration coincide.
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 an 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 a given 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 is equal to the sum of the moments of all forces acting in the system in relation 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 rigid 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. The 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?
Definition 1
The moment of force is represented by a torque or rotational moment, being a vector physical quantity.
It is defined as the vector product of the force vector, as well as 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 a force on a solid 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 internal force that arises 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 in the form of the so-called “rotational force”. The SI unit of measurement is the newton meter. The moment of a force may also be called the "moment of a couple of forces", as noted in Archimedes' work on levers.
Note 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 vector product format:
$\vec (M)=\vec(r)\vec(F)$, 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 torque would 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 energy of 2 $\pi$ joules. Mathematically it looks like this:
$E = M\theta$, where:
- $E$ represents energy;
- $M$ is considered to be the torque;
- $\theta$ will be the angle in radians.
Today, the measurement of moment of force is carried out by using special load sensors of strain gauge, optical and inductive types.
Formulas for calculating moment of force
An interesting thing in physics is the calculation of the moment of force in a field, produced according to the formula:
$\vec(M) = \vec(M_1)\vec(F)$, where:
- $\vec(M_1)$ is considered the lever moment;
- $\vec(F)$ 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. If 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
Note 2
When a body is exposed to two equal and also oppositely directed forces, which do not lie on the same straight line, the absence of this body in a state of equilibrium is observed. This is explained by the fact that the resulting moment of the indicated forces relative to any of the axes does not have a zero value, since both represented forces have moments directed in the same direction (a pair of forces).
In a situation where the body is fixed on an axis, it will rotate under the influence of a couple of forces. If a pair of forces is applied to a free body, it will then 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 (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 moment of several forces is equal to zero, it will be considered the same relative to all axes parallel to each other. For this reason, the effect on the body of all these forces can be replaced by the action of just one pair of forces with the same moment.
A moment of power relative to an arbitrary center in the plane of action of the force, the product of the force modulus and the shoulder 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, counterclockwise - plus;
The moment of force can be expressed as a vector. This is 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 systems of forces.
Let's consider the moment of force about the axis, calculate the moment of force about the Z axis;
Let's 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 relative to 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 intersects it, then m z (F)=0
Expressing moment of force as a vector expression
Let's draw r a to point A. Consider OA x F.
This is the third vector m o , perpendicular to the plane. The magnitude of the cross product can be calculated using twice the area of the shaded triangle.
Analytical expression of force relative to coordinate axes.
Let us assume that the Y and Z, X axes with unit vectors i, j, k are associated with 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 a 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 connections. To do this, the basic equations in a plane system of forces are used.
2 or 3 moment equations can be used.
Example
Let's create an equation for the sum of all forces on the X and Y axis:
The sum of the moments of all forces relative to point A:
Parallel forces
Equation for point A:
Equation for point B:
The sum of the projections of forces on the Y axis.
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 arm, the less force must be applied to obtain the 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 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 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 pairs is always equal to the product of 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.