Rotation is a typical type of mechanical movement that is often found in nature and technology. Any rotation occurs as a result of the influence 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 us characterize the systems to which this concept can be applied. A rotation system presupposes the presence of an axis around which circular motion 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, and 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.
Determination of moment of force
Now let's move on to the immediate topic of the article. According to the physical definition, it is the vector product of the vector of application of force 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 formula for the torque M¯, the force F¯ can be directed in any way relative to the direction of the axis. However, a force component parallel to the axis will not produce rotation if the axis is rigidly fixed. In most problems in physics, one has to consider forces F¯, which lie in planes perpendicular to the axis of rotation. In these cases, the absolute value of the torque can be determined using the following formula:
|M¯| = |r¯|*|F¯|*sin(β).
Where β is the angle between the vectors r¯ and F¯.
What is leverage?
The force lever plays an important role in determining the magnitude of the moment of force. To understand what we are talking about, consider the following figure.
Shown here is a rod of length L, which is fixed at the point of rotation 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 moment of force, we can write:
M = 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. Taking into account the periodicity of the trigonometric sine function, we can rewrite this equality as follows:
Now let's turn our attention to a right triangle built on sides L, d and F. By the definition of the sine function, the product of the hypotenuse L and the sine of the angle φ gives the value of leg d. Then we come to equality:
The linear quantity 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, the concept of a force lever is convenient to use when calculating the moment M. The resulting formula says that the maximum torque for a certain force F will occur only when the length of the radius vector r¯ (L¯ in the figure above) is equal to lever of force, 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? Answering this question is not particularly difficult if we remember that the result of the product of two vectors is a third vector, which lies on an axis perpendicular to the plane of location of the original vectors.
It remains to decide whether the moment of force will be directed upward or downward (toward or away from the reader) relative to the mentioned plane. This can be determined either by the gimlet rule or by the right hand rule. Here are both rules:
- Right hand rule. If you position the right hand 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 point in 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¯. Remember that it only rotates clockwise.
Both rules are equal, so everyone can use the one that is more convenient for them.
When solving practical problems, different directions of torque (up - down, left - right) are taken into account using the "+" or "-" signs. It should be remembered that the positive direction of the moment M¯ is considered to be one that leads to the rotation of the system counterclockwise. Accordingly, if a certain force causes the system to rotate in the direction of the clock, then the moment it creates will have a negative value.
Physical meaning of the quantity M¯
In the physics and mechanics of rotation, the value M¯ determines the ability of a force or a sum of forces to perform rotation. Since the mathematical definition of the value M¯ includes not only the 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 kind of ability we are talking about, here are a few examples:
- Every person, at least once in his life, tried to open a door, not by grasping 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 a bolt, use special wrenches. The longer the wrench, the easier it is to unscrew the nut.
- To feel the importance of the lever of force, we invite readers to do the following experiment: take a chair and try to hold it suspended with one hand, in one case lean your hand against your body, in another - perform the task with a straight arm. The latter will be an impossible task for many, although the weight of the chair remains the same.
Torque units
A few words should also be said about the SI units in which torque is measured. According to the formula written down 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.
However, the coincidence of units of moment of force with units of energy is not accidental. The work done to rotate the system, performed by the moment M, is calculated by the formula:
From this we find that M can also be expressed in joules per radian (J/rad).
Dynamics of rotation
At the beginning of the article, we wrote down the kinematic characteristics that are used to describe the rotational motion. 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 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 up in a known time t.
Example of problem solution
The weightless lever, which is 2 meters long, has a support in the middle. What weight should be placed on one end of the lever so that it is in a state of equilibrium if a load weighing 10 kg lies on the other side of the support at a distance of 0.5 meters from it?
Obviously, what will happen if the moments of force created by the loads are equal in magnitude. The force creating the moment in this problem is the weight of the body. The levers of force are equal to the distances from the loads to the support. Let us write the corresponding equality:
m 1 *g*d 1 = m 2 *g*d 2 =>
P 2 = m 2 *g = m 1 *g*d 1 /d 2 .
We obtain the weight P 2 if we substitute from the problem conditions 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 newton.
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 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. 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?
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.
In the article we will talk about the moment of force about a point and an axis, definitions, drawings and graphs, what unit of measurement of the moment of force, work and force in rotational motion, as well as examples and problems.
Moment of power represents a vector of a physical quantity equal to the product of vectors shoulder strength(radius vector of the particle) and strength, acting on a point. The force lever is a vector connecting the point through which the axis of rotation of a rigid body passes with the point to which the force is applied.
where: r is the force arm, F is the force applied to the body.
Vector direction moment forces always perpendicular to the plane defined by the vectors r and F.
Main point- any system of forces on a plane relative to the accepted pole is called the algebraic moment of the moment of all forces of this system relative to this pole.
In rotational movements, not only the physical quantities themselves are important, but also how they are located relative to the axis of rotation, that is, their moments. We already know that in rotational motion, not only mass is important, but also. In the case of a force, its effectiveness in triggering acceleration is determined by the way that force is applied to the axis of rotation.
The relationship between force and the way it is applied describes MOMENT OF POWER. The moment of force is the vector product of the force arm R to the force vector F:
As in every vector product, so here
Therefore, the force will not affect the rotation when the angle between the force vectors F and lever R equal to 0 o or 180 o. What is the effect of applying a moment of force M?
We use Newton's Second Law of Motion and the relationship between rope and angular velocity v = Rω in scalar form, are valid when the vectors R And ω perpendicular to each other
Multiplying both sides of the equation by R, we get
Since mR 2 = I, we conclude that
The above dependence is also valid for the case of a material body. Note that while the external force gives a linear acceleration a, the moment of the external force gives the angular acceleration ε.
Unit of measurement of moment of force
The main measure of the moment of force in the SI system coordinate is: [M]=N m
In GHS: [M]=din cm
Work and force in rotational motion
Work in linear motion is determined by the general expression,
but in rotational motion,
and consequently
Based on the properties of the mixed product of three vectors, we can write
Therefore we have obtained an expression for work in rotational motion:
Power in rotational motion:
Find moment of power, acting on the body in the situations shown in the figures below. Let's assume that r = 1m and F = 2N.
A) since the angle between the vectors r and F is 90°, then sin(a)=1:
M = r F = 1m 2N = 2N m
b) because the angle between vectors r and F is 0°, so sin(a)=0:
M = 0
yes directed force can't give a point rotational movement.
c) since the angle between vectors r and F is 30°, then sin(a)=0.5:
M = 0.5 r F = 1 N m.
Thus, the directed force will cause body rotation, however, its effect will be less than in the case a).
Moment of force about the axis
Let's assume the data is a point O(pole) and power P. At the point O we take the origin of a rectangular coordinate system. Moment of power R in relation to the poles O represents a vector M from (R), (picture below) .
Any point A on line P
has coordinates (xo, yo, zo).
Force vector P
has coordinates Px, Py, Pz.
Combining point A (xo, yo, zo) with the beginning of the system, we get the vector p. Force vector coordinates P
relative to the pole O indicated by symbols Mx, My, Mz.
These coordinates can be calculated as the minima of a given determinant, where ( i, j, k) - unit vectors on the coordinate axes (options): i, j, k
After solving the determinant, the coordinates of the moment will be equal to:
Moment vector coordinates Mo (P) are called moments of force about the corresponding axis. For example, moment of force P relative to the axis Oz surrounds template:
Mz = Pyxo - Pxyo
This pattern is interpreted geometrically as shown in the figure below.
Based on this interpretation, the moment of force about the axis Oz can be defined as the moment of force projection P
perpendicular to the axis Oz relative to the point of penetration of this plane by the axis. Projection of force P
perpendicular to the axis is indicated Pxy
, and the plane penetration point Oxy- axis OS symbol O.
From the above definition of the moment of a force about an axis, it follows that the moment of a force about an axis is zero when the force and the axis are equal, in the same plane (when the force is parallel to the axis or when the force intersects the axis).
Using the formulas on Mx, My, Mz,
we can calculate the value of the moment of force P
relative to the point O and determine the angles contained between the vector M
and system axes:
Torque mark:
plus (+) - rotation of the force around the O axis clockwise,
minus (-) — rotation of the force around the O axis counterclockwise.
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.
