Conditions for the equilibrium of bodies. Formula of the resultant of all forces Definition and formula of the resultant of all forces
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TwitterStatics is a branch of mechanics that studies the conditions of equilibrium of bodies.
It follows from Newton's second law that if the geometric sum of all external forces applied to a body is zero, then the body is at rest or performs uniform rectilinear motion. In this case, it is customary to say that the forces applied to the body balance each other. When calculating resultant all forces acting on a body can be applied to center of gravity .
For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero.
On fig. 1.14.1 gives an example of the equilibrium of a rigid body under the action of three forces. Intersection point O lines of action of forces and does not coincide with the point of application of gravity (center of mass C), but at equilibrium these points are necessarily on the same vertical. When calculating the resultant, all forces are reduced to one point.
If the body can rotate about some axis, then for its equilibrium it is not enough to equal zero the resultant of all forces.
The rotating action of a force depends not only on its magnitude, but also on the distance between the line of action of the force and the axis of rotation.
The length of the perpendicular drawn from the axis of rotation to the line of action of the force is called shoulder of strength.
The product of the modulus of force per shoulder d called moment of force M. The moments of those forces that tend to rotate the body counterclockwise are considered positive (Fig. 1.14.2).
moment rule : a body with a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body about this axis is zero:
In the International System of Units (SI), moments of forces are measured in Hnewton— meters (N∙m) .
In the general case, when a body can move translationally and rotate, both conditions must be met for equilibrium: the resultant force must be equal to zero and the sum of all moments of forces must be equal to zero.
Wheel rolling on a horizontal surface - example indifferent balance(Fig. 1.14.3). If the wheel is stopped at any point, it will be in equilibrium. Along with indifferent equilibrium in mechanics, states are distinguished sustainable and unstable balance.
A state of equilibrium is called stable if, with small deviations of the body from this state, forces or moments of forces arise that tend to return the body to an equilibrium state.
With a small deviation of the body from the state of unstable equilibrium, forces or moments of forces arise that tend to remove the body from the equilibrium position.
A ball lying on a flat horizontal surface is in a state of indifferent equilibrium. A ball located at the top of a spherical ledge is an example of an unstable equilibrium. Finally, the ball at the bottom of the spherical cavity is in a state of stable equilibrium (Fig. 1.14.4).
For a body with a fixed axis of rotation, all three types of equilibrium are possible. Indifferent equilibrium occurs when the axis of rotation passes through the center of mass. In stable and unstable equilibrium, the center of mass is on a vertical line passing through the axis of rotation. In this case, if the center of mass is below the axis of rotation, the state of equilibrium is stable. If the center of mass is located above the axis, the equilibrium state is unstable (Fig. 1.14.5).
A special case is the equilibrium of a body on a support. In this case, the elastic force of the support is not applied to one point, but is distributed over the base of the body. A body is in equilibrium if a vertical line drawn through the center of mass of the body passes through footprint, i.e., inside the contour formed by lines connecting the support points. If this line does not cross the area of support, then the body overturns. An interesting example of the equilibrium of a body on a support is the leaning tower in the Italian city of Pisa (Fig. 1.14.6), which, according to legend, was used by Galileo when studying the laws of free fall of bodies. The tower has the shape of a cylinder with a height of 55 m and a radius of 7 m. The top of the tower deviates from the vertical by 4.5 m.
A vertical line drawn through the center of mass of the tower intersects the base approximately 2.3 m from its center. Thus, the tower is in a state of equilibrium. The balance will be disturbed and the tower will fall when the deviation of its top from the vertical reaches 14 m. Apparently, this will not happen very soon.
Systematization of knowledge about the resultant of all forces applied to the body; about vector addition.
Lesson objectives (for teacher):
Educational:
- Clarify and expand knowledge about the resultant force and how to find it.
- To form the ability to apply the concept of the resultant force to the justification of the laws of motion (Newton's laws)
- Determine the level of mastering the topic;
- Continue to develop the skills of self-analysis of the situation and self-control.
Educational:
- To contribute to the formation of the worldview idea of the cognizability of phenomena and properties of the surrounding world;
- Emphasize the importance of modulation in the cognizability of matter;
- Pay attention to the formation of universal human qualities:
a) efficiency,
b) independence;
c) accuracy;
d) discipline;
e) responsible attitude to learning.
Developing:
a) highlight signs of similarity in the description of phenomena,
b) analyze the situation
c) make logical inferences based on this analysis and existing knowledge;
Equipment and demonstrations.
- Illustrations:
sketch for the fable by I.A. Krylov "Swan, crayfish and pike",
sketch of the painting by I. Repin “Barge haulers on the Volga”,
to problem No. 108 “Turnip” - “Physicist's Task Book” by G. Oster. - Arrows colored on a polyethylene basis.
- Copy paper.
- Kodoscope and film with the solution of two problems of independent work.
- Shatalov "Supporting notes".
- Faraday's portrait.
Board layout:
“If you are in this
figure it out properly
you better be able to follow
following my train of thought
in what follows."
M. Faraday
During the classes
1. Organizational moment
Examination:
- absent;
- the presence of diaries, notebooks, pens, rulers, pencils;
Appearance rating.
2. Repetition
As we talk in class, we repeat:
- I Newton's law.
- Force is the cause of acceleration.
- Newton's second law.
- Addition of vectors to the rule of a triangle and a parallelogram.
3. Main material
Lesson problem.
“Once a Swan, Cancer and Pike
Carried with luggage, a cart came from
And together, three, all harnessed to it;
Out of the skin climb out
And the cart still does not move!
The luggage would have seemed easy for them:
Yes, the swan breaks into the clouds,
Cancer moves back
And Pike pulls into the water!
Who is guilty of them, who is right -
It is not for us to judge;
Yes, only things are still there!”(I.A. Krylov)
The fable expresses a skeptical attitude towards Alexander I, it ridicules the troubles in the State Council of 1816, the reforms and committees started by Alexander I were unable to move the deeply bogged down cart of autocracy. In this, from a political point of view, Ivan Andreevich was right. But let's find out the physical aspect. Is Krylov right? To do this, it is necessary to become more familiar with the concept of the resultant of forces applied to the body.
A force equal to the geometric sum of all forces applied to the body (point) is called the resultant or resultant force.
Picture 1
How does this body behave? Either it is at rest or it moves in a straight line and uniformly, since it follows from Newton's I law that there are such frames of reference with respect to which a progressively moving body retains its speed constant if no other bodies act on it or the action of these bodies is compensated,
i.e. |F 1 | = |F 2 | (the definition of the resultant is introduced).
A force that produces the same effect on a body as several simultaneously acting forces is called the resultant of these forces.
Finding the resultant of several forces is the geometric addition of the acting forces; is carried out according to the rule of a triangle or parallelogram.
In figure 1 R=0, because .
To add two vectors, the beginning of the second is applied to the end of the first vector and the beginning of the first is connected to the end of the second (manipulation on a board with polyethylene-based arrows). This vector is the resultant of all forces applied to the body, i.e. R \u003d F 1 - F 2 \u003d 0
How can one formulate Newton's first law based on the definition of the resultant force? The well-known formulation of Newton's first law:
“If other bodies do not act on a given body or the actions of other bodies are compensated (balanced), then this body is either at rest or moves in a straight line and uniformly.”
New formulation of Newton's I law (give the formulation of Newton's I law for the record):
“If the resultant of the forces applied to the body is zero, then the body retains its state of rest or uniform rectilinear motion.”
How to proceed when finding the resultant, if the forces applied to the body are directed in one direction along one straight line?
Task #1 (solution of problem No. 108 by Grigory Oster from the problem book “Physics”).
The grandfather, holding the turnip, develops a traction force up to 600 N, the grandmother - up to 100 N, the granddaughter - up to 50 N, the Bug - up to 30 N, the cat - up to 10 N and the mouse - up to 2 N. What is the resultant of all these forces, pointing in the same straight line in the same direction? Would this company handle the turnip without a mouse if the forces holding the turnip in the ground are 791 N?
(Manipulation on a board with polyethylene-based arrows).
Answer. The module of the resultant force, equal to the sum of the modules of forces with which the grandfather pulls the turnip, the grandmother pulls the grandfather, the granddaughter pulls the grandmother, the Bug pulls the granddaughter, the cat pulls the Bug, and the mouse pulls the cat, will be equal to 792 N. The contribution of the muscle force of the mouse to this mighty impulse is 2 N. Without Myshkin's Newtons, things will not work.
Task number 2.
And if the forces acting on the body are directed at right angles to each other? (Manipulation on a board with polyethylene-based arrows).
(We write down the rules p. 104 Shatalov “Support notes”).
Task number 3.
Let's try to find out if I.A. is right in the fable. Krylov.
If we assume that the traction force of the three animals described in the fable is the same and comparable (or more) with the weight of the cart, and also exceeds the static friction force, then, using Figure 2 (1) for Problem 3, we obtain after constructing the resultant that And .BUT. Krylov, of course, is right.
If we use the data below, prepared by the students in advance, then we get a slightly different result (see Figure 2 (1) for task 3).
| Name | Dimensions, cm | Weight, kg | Speed, m/s |
| Cancer (river) | 0,2 - 0,5 | 0,3 - 0,5 | |
| Pike | 60 -70 | 3,5 – 5,5 | 8,3 |
| Swan | 180 | 7 – 10 (13) | 13,9 – 22,2 |
The power developed by bodies during uniform rectilinear motion, which is possible if the traction force and the resistance force are equal, can be calculated using the following formula.
Newton's laws are a mathematical abstraction. In reality, the cause of the movement or rest of bodies, as well as their deformation, are several forces at once. Therefore, an important addition to the laws of mechanics will be the introduction of the concept of the resultant force and its application.
About the reasons for the changes
Classical mechanics is divided into two sections - kinematics, which describes the trajectory of motion of bodies with the help of equations, and dynamics, which deals with the reasons for changing the position of objects or the objects themselves.
The reason for the changes is a certain force, which is a measure of the action of other bodies or force fields on the body (for example, an electromagnetic field or gravity). For example, the force of elasticity causes deformation of the body, the force of gravity - the fall of bodies to the Earth.
Force is a vector quantity, that is, its action is directed. The modulus of force is generally proportional to a certain coefficient (for the deformation of a spring, this is its stiffness), as well as to the parameters of action (mass, charge).
For example, in the case of the Coulomb force, this is the magnitude of both charges, taken modulo, the squared distance between the charges and the coefficient k, in the SI system, defined by the expression: $k = (1 \over 4 \pi \epsilon)$, where $\epsilon$ is the dielectric constant.
Addition of forces
In the case when n forces act on the body, they speak of the resultant force, and the formula for Newton's second law takes the form:
$m\vec a = \sum\limits_(i=1)^n \vec F_i$.
Rice. 1. The resultant of forces.
Since F is a vector quantity, the sum of forces is called geometric (or vector). Such addition is performed according to the rule of a triangle or parallelogram, or by components. Let's explain each method with an example. To do this, we write the formula for the resultant force in a general form:
$F = \sum\limits_(i=1)^n \vec F_i$
And the force $F_i$ can be represented as:
$F = (F_(xi), F_(yi), F_(zi))$
Then the sum of the two forces will be the new vector $F_(ab) = (F_(xb) + F_(xa), F_(yb) + F_(ya), F_(zb) + F_(za))$.
Rice. 2. Componentwise addition of vectors.
The absolute value of the resultant can be calculated as follows:
$F = \sqrt((F_(xb) + F_(xa))^2 + (F_(yb) + F_(ya))^2 + (F_(zb) + F_(za))^2)$
Now let's give a strict definition: the resultant force is the vector sum of all the forces that affect the body.
Let's analyze the rules of a triangle and a parallelogram. Graphically it looks like this:
Rice. 3. Rule of triangle and parallelogram.
Outwardly, they seem different, but when it comes to calculations, they come down to finding the third side of a triangle (or, which is the same thing, the diagonal of a parallelogram) using the cosine theorem.
If there are more than two forces, it is sometimes more convenient to use the polygon rule. At its core, this is still the same triangle, only repeated in one figure a certain number of times. If, as a result, the contour turned out to be closed, the total action of forces is equal to zero and the body is at rest.
Tasks
- A box placed in the center of a Cartesian rectangular coordinate system is subject to two forces: $F_1 = (5, 0)$ and $F_2 = (3, 3)$. Calculate the resultant in two ways: according to the triangle rule and using component-by-component addition of vectors.
Decision
The resultant force will be the vector sum of $F_1$ and $F_2$.
Therefore, we write:
$\vec F = \vec F_1 + \vec F_2 = (5+3, 0+3) = (8, 3)$
The absolute value of the resultant force:
$F = \sqrt(8^2 + 3^2) = \sqrt(64 + 9) = 8.5 N$
Now we get the same value using the triangle rule. To do this, we first find the absolute values of $F_1$ and $F_2$, as well as the angle between them.
$F_1 = \sqrt(5^2 + 0^2) = 5 N$
$F_2 = \sqrt(3^2 + 3^2) = 4.2 N$
The angle between them is 45˚, since the first force is parallel to the Ox axis, and the second divides the first coordinate plane in half, that is, it is the bisector of a right angle.
Now, having placed the vectors according to the triangle rule, we calculate the resultant using the cosine theorem:
$F = \sqrt(F_1^2 + F_2^2 - 2F_1F_2 cos135) = \sqrt(F_1^2 + F_2^2 + 2F_1F_2 sin45) = \sqrt(25 + 18 + 2 \cdot 5 \cdot 4,2 \ cdot sin45) = 8.5 N$
- Three forces act on the machine: $F_1 = (-5, 0)$, $F_2 = (-2, 0)$, $F_1 = (7,0)$. What is their resultant?
Decision
It is enough to add the x components of the vectors:
$F = -5 - 2 + 7 = 0$
What have we learned?
During the lesson, the concept of the resultant of forces was introduced and various methods for its calculation were considered, as well as Newton's second law was introduced for the general case when the number of forces is unlimited.
Topic quiz
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In inertial reference systems, a change in the speed of a body is possible only when another body acts on it. Quantitatively, the action of one body on another is expressed using such a physical quantity as force (). The impact of one body on another can cause a change in the speed of the body, both in magnitude and in direction. Therefore, the force is a vector and is determined not only by the magnitude (modulus), but also by the direction. The direction of the force determines the direction of the acceleration vector of the body affected by the force in question.
The magnitude and direction of force is determined by Newton's second law:
where m is the mass of the body on which the force acts - the acceleration that the force imparts to the body in question. The meaning of Newton's second law lies in the fact that the forces that act on the body determine how the speed of the body changes, and not just its speed. Note that Newton's second law is valid only in inertial frames of reference.
If several forces act simultaneously on the body, then the body moves with an acceleration that is equal to the vector sum of the accelerations that would appear under the influence of each of the bodies separately. The forces acting on the body and applied to its one point should be added in accordance with the rule of vector addition.
DEFINITION
The vector sum of all forces acting on the body at the same time is called resultant force ():
If several forces act on the body, then Newton's second law is written as:
The resultant of all forces acting on the body can be equal to zero if there is a mutual compensation of the forces applied to the body. In this case, the body moves at a constant speed or is at rest.
When depicting the forces acting on the body, in the drawing, in the case of a uniformly accelerated movement of the body, the resultant force directed along the acceleration should be depicted longer than the oppositely directed force (the sum of forces). In the case of uniform motion (or rest), the dyne of force vectors directed in opposite directions is the same.
To find the resultant force, it is necessary to depict on the drawing all the forces that must be taken into account in the problem acting on the body. The forces must be added according to the rules of vector addition.
Examples of problem solving
EXAMPLE 1
| Exercise | The body rests on an inclined plane (Fig. 1), depict the forces that act on the body, what is the resultant of all forces applied to the body? |
| Decision | Let's make a drawing. On a body located on an inclined plane, the force of gravity (), the force of the normal reaction of the support () and the force of static friction (according to the condition, the body does not move) () act. The resultant of all forces acting on the body () can be found by vector summation: We first add, according to the parallelogram rule, the force of gravity and the reaction force of the support, we get the force. This force must be directed along the inclined plane along the movement of the body. The length of the vector must be equal to the thorn force vector, since the body is at rest according to the condition. According to Newton's second law, the resultant must be zero: |
| Answer | The resultant force is zero. |
EXAMPLE 2
| Exercise | A load suspended in air on a spring moves with constant downward acceleration (Fig. 3), what forces act on the load? What is the resultant force applied to the load? Where will the resultant force be directed? |
| Decision | Let's make a drawing. The load suspended on the spring is affected by: the force of gravity () from the side of the Earth and the elastic force of the spring () (from the side of the spring), when the load moves in the air, usually the force of friction of the load against the air is neglected. The resultant of the forces applied to the load in our problem can be found as: |
This is the vector sum of all the forces acting on the body.
The cyclist leans towards the turn. The force of gravity and the reaction force of the support from the ground give the resultant force that imparts the centripetal acceleration necessary for movement in a circle
Relationship with Newton's second law
Let's remember Newton's law: 
The resultant force can be equal to zero in the case when one force is compensated by another, the same force, but opposite in direction. In this case, the body is at rest or moving uniformly.
If the resultant force is NOT equal to zero, then the body moves with uniform acceleration. Actually, it is this force that is the cause of uneven movement. Direction of the resultant force always coincides in direction with the acceleration vector.
When it is required to depict the forces acting on the body, while the body moves uniformly accelerated, it means that in the direction of acceleration the acting force is longer than the opposite one. If the body moves uniformly or is at rest, the length of the force vectors is the same.
Finding the resultant force
In order to find the resultant force, it is necessary: firstly, to correctly designate all the forces acting on the body; then draw coordinate axes, choose their directions; at the third step, it is necessary to determine the projections of the vectors on the axes; write equations. Briefly: 1) designate the forces; 2) choose axes, their directions; 3) find the projections of forces on the axis; 4) write down the equations.
How to write equations? If the body moves uniformly in some direction or is at rest, then the algebraic sum (taking into account the signs) of the force projections is equal to zero. If a body moves uniformly accelerated in a certain direction, then the algebraic sum of the projections of forces is equal to the product of mass and acceleration, according to Newton's second law.
Examples
A body moving uniformly on a horizontal surface is affected by the force of gravity, the reaction force of the support, the force of friction and the force under which the body moves.
We denote the forces, choose the coordinate axes
Let's find projections
Writing down the equations
A body that is pressed against a vertical wall moves downward with uniform acceleration. The body is affected by gravity, friction, support reaction and the force with which the body is pressed. The acceleration vector is directed vertically downwards. The resultant force is directed vertically downward.
The body moves uniformly along the wedge, the slope of which is alpha. The force of gravity, the reaction force of the support, and the force of friction act on the body.
The main thing to remember
1) If the body is at rest or moves uniformly, then the resultant force is zero and the acceleration is zero;
2) If the body moves uniformly accelerated, then the resultant force is not zero;
3) The direction of the resultant force vector always coincides with the direction of acceleration;
4) Be able to write down the equations of the projections of the forces acting on the body
Block - a mechanical device, a wheel rotating around its axis. Blocks can be mobile and motionless.
Fixed block used only to change the direction of the force.
Bodies connected by an inextensible thread have the same accelerations.
Movable block designed to change the amount of effort applied. If the ends of the rope wrapping around the block make equal angles with the horizon, then a force half as much as the weight of the load will be required to lift the load. The force acting on the load is related to its weight, as the radius of the block is to the chord of the arc wrapped around the rope.
The acceleration of body A is half that of body B.
In fact, every block is lever arm, in the case of a fixed block - equal arms, in the case of a movable block - with a shoulder ratio of 1 to 2. As for any other lever, the rule is true for the block: how many times we win in effort, how many times we lose in distance
A system consisting of a combination of several movable and fixed blocks is also used. Such a system is called a polyspast.
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