The resultant of all forces is zero. Conditions for equilibrium of bodies. Finding the resultant force
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 develop the ability to apply the concept of resultant force to substantiate the laws of motion (Newton’s laws)
- Identify the level of mastery of the topic;
- Continue developing the skills of self-analysis of the situation and self-control.
Educational:
- To promote the formation of a worldview idea of the knowability of phenomena and properties of the surrounding world;
- Emphasize the importance of modulation in the cognition of matter;
- Pay attention to the formation of universal human qualities:
a) efficiency,
b) independence;
c) accuracy;
d) discipline;
e) responsible attitude towards learning.
Educational:
a) highlight signs of similarity in the description of phenomena,
b) analyze the situation
c) draw logical conclusions 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 I. Repin’s painting “Barge Haulers on the Volga”,
for problem No. 108 “Turnip” - “Physics Problem Book” by G. Oster. - Colored arrows on a polyethylene base.
- Copy paper.
- An overhead projector and film with a solution to two independent work problems.
- Shatalov “Supporting notes”.
- Portrait of Faraday.
Board design:
“If you're into this
figure it out properly
you'll be able to keep track better
following my train of thought
when presenting what follows.”
M. Faraday
During the classes
1. Organizational moment
Examination:
- absent;
- availability of diaries, notebooks, pens, rulers, pencils;
Appearance assessment.
2. Repetition
During the conversation in class we repeat:
- Newton's first law.
- Force is the cause of acceleration.
- Newton's II law.
- Addition of vectors according to the triangle and parallelogram rule.
3. Main material
Lesson problem.
“Once upon a time a Swan, a Crayfish and a Pike
They began to carry a load of luggage
And together, the three of them, all harnessed themselves to it;
They're going out of their way to
But the cart still doesn’t move!
The luggage would seem light to them:
Yes, the Swan rushes into the clouds,
Cancer is moving backwards
And the Pike is pulling into the water!
Who is to blame and who is right?
It is not for us to judge;
But the cart is 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 initiated by Alexander I were unable to move the deeply bogged cart of autocracy. In this, from a political point of view, Ivan Andreevich was right. But let's figure 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 a body.
A force equal to the geometric sum of all forces applied to a body (point) is called the resultant or resultant force.
Picture 1
How does this body behave? Either it is at rest or it moves rectilinearly and uniformly, since from Newton’s First Law it follows that there are such reference systems relative to which a translationally moving body maintains its speed constant if other bodies do not 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; performed according to the triangle or parallelogram rule.
In Figure 1 R=0, because .
To add two vectors, apply the beginning of the second to the end of the first vector and connect the beginning of the first to the end of the second (manipulation on a board with arrows on a polyethylene base). This vector is the resultant of all forces applied to the body, i.e. R = F 1 – F 2 = 0
How can we formulate Newton’s First Law based on the definition of the resultant force? The already known formulation of Newton's First Law:
“If a given body is not acted upon by other bodies or the actions of other bodies are compensated (balanced), then this body is either at rest or moving rectilinearly and uniformly.”
New formulation of Newton's first law (give the formulation of Newton’s First Law for the record):
“If the resultant of the forces applied to the body is equal to zero, then the body maintains its state of rest or uniform rectilinear motion.”
What to do when finding the resultant if the forces applied to the body are directed in one direction along one straight line?
Task No. 1 (solution to problem No. 108 by Grigory Oster from the Physics problem book).
Grandfather, holding a turnip, develops a traction force of up to 600 N, grandmother - up to 100 N, granddaughter - up to 50 N, Bug - up to 30 N, cat - up to 10 N and mouse - up to 2 N. What is the resultant of all these forces? directed in one straight line in the same direction? Could this company handle the turnip without a mouse if the forces holding the turnip in the ground are equal to 791 N?
(Manipulation on a board with arrows on a polyethylene base).
Answer. The modulus of the resultant force, equal to the sum of the moduli of forces with which the grandfather pulls the turnip, the grandmother for the grandfather, the granddaughter for the grandmother, the Bug for the granddaughter, the cat for the Bug, and the mouse for the cat, will be equal to 792 N. The contribution of the muscular force of the mouse to this powerful impulse is equal to 2 N. Without Myshkin’s newtons, things won’t work.
Task No. 2.
What if the forces acting on the body are directed at right angles to each other? (Manipulation on a board with arrows on a polyethylene base).
(We write down the rules p. 104 Shatalov “Basic notes”).
Task No. 3.
Let's try to find out whether 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, after constructing the resultant, we obtain that And .A. Krylov is certainly right.
If we use the data below, prepared by students in advance, we get a slightly different result (see Figure 2 (1) for task 3).
| Name | Dimensions, cm | Weight, kg | Speed, m/s |
| Crayfish (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 when the traction force and resistance force are equal, can be calculated using the following formula.
This vector sum all forces acting on the body.
The cyclist leans towards the turn. The force of gravity and the reaction force of the support from the earth provide a resultant force that imparts the centripetal acceleration necessary for motion 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 zero, then the body moves uniformly accelerated. Actually, it is this force that causes the uneven movement. Direction of resultant force Always coincides in direction with the acceleration vector.
When it is necessary to depict the forces acting on a body, while the body moves with uniform acceleration, 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, true designate all the forces, acting on the body; then draw coordinate axes, choose their directions; in the third step it is necessary to determine projections vectors on the axis; write down the equations. Briefly: 1) identify the forces; 2) select the axes and their directions; 3) find the projections of forces on the axis; 4) write down the equations.
How to write equations? If in a certain direction the body moves uniformly or is at rest, then the algebraic sum (taking into account signs) of the projections of forces 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 subject to the force of gravity, the reaction force of the support, the force of friction and the force under which the body moves.
Let us denote the forces, choose the coordinate axes
Let's find the projections
Writing down the equations
A body that is pressed against a vertical wall moves downward with uniform acceleration. The body is acted upon by the force of gravity, the force of friction, the reaction of the support and the force with which the body is pressed. The acceleration vector is directed vertically downwards. The resultant force is directed vertically downwards.
The body moves uniformly along a wedge whose slope is alpha. The body is acted upon by the force of gravity, the reaction force of the support, and the force of friction.
The main thing to remember
1) If the body is at rest or moving 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 equations of projections of forces acting on a body
A block is a mechanical device, a wheel that rotates around its axis. Blocks can be mobile And motionless.
Fixed block used only to change the direction of force.
Bodies connected by an inextensible thread have equal accelerations.
Movable block designed to change the amount of effort applied. If the ends of the rope clasping the block make equal angles with the horizon, then lifting the load will require a force half as much as the weight of the load. The force acting on a load is related to its weight as the radius of a block is to the chord of an arc encircled by a rope.
The acceleration of body A is half the acceleration of body B.
In fact, any block is lever arm, in the case of a fixed block - equal arms, in the case of a movable one - with a ratio of shoulders of 1 to 2. As for any other lever, the following rule applies to the block: the number of times we win in effort, the same number of times we lose in distance
A system consisting of a combination of several movable and fixed blocks is also used. This system is called a polyspast.
Igor Babin (St. Petersburg) 14.05.2012 17:33
The condition says that you need to find the weight of the body.
and in the solution the modulus of gravity.
How can weight be measured in Newtons?
There is an error in the condition(
Alexey (St. Petersburg)
Good afternoon
You are confusing the concepts of mass and weight. The weight of a body is the force (and therefore weight is measured in Newtons) with which the body presses on a support or stretches a suspension. As follows from the definition, this force is applied not even to the body, but to the support. Weightlessness is a state when a body loses not mass, but weight, that is, the body stops putting pressure on other bodies.
I agree that the decision took some liberties in the definitions, which have now been corrected.
Yuri Shoitov (Kursk) 26.06.2012 21:20
The concept of “body weight” was introduced into educational physics extremely unsuccessfully. If in the everyday concept weight means mass, then in school physics, as you correctly noted, the weight of a body is the force (and therefore weight is measured in Newtons) with which the body presses on a support or stretches a suspension. Note that we are talking about one support and one thread. If there are several supports or threads, the concept of weight disappears.
Let me give you an example. Let a body be suspended in a liquid by a thread. It stretches the thread and presses on the liquid with a force equal to minus the Archimedes force. Why, when talking about the weight of a body in a liquid, do we not add up these forces, as you do in your solution?
I registered on your site, but did not notice what had changed in our communication. Please excuse my stupidity, but being an old man, I am not fluent enough to navigate the site.
Alexey (St. Petersburg)
Good afternoon
Indeed, the concept of body weight is very vague when the body has several supports. Typically, the weight in this case is defined as the sum of interactions with all supports. In this case, the impact on gaseous and liquid media is, as a rule, excluded. This exactly falls under the example you described, with a weight suspended in water.
Here I immediately remember a children's problem: “What weighs more: a kilogram of fluff or a kilogram of lead?” If we solve this problem honestly, then we must undoubtedly take into account the power of Archimedes. And by weight, most likely, we will understand what the scales will show us, that is, the force with which fluff and lead press, say, on the scales. That is, here the force of interaction with air is, as it were, excluded from the concept of weight.
On the other hand, if we assume that we have pumped out all the air and put a body on the scales to which a string is attached. Then the force of gravity will be balanced by the sum of the reaction force of the support and the tension force of the thread. If we understand weight as the force acting on supports that prevent a fall, then the weight here will be equal to this sum of the tensile force of the thread and the force of pressure on the scale, that is, the same in magnitude as the force of gravity. The question arises again: is the thread better or worse than Archimedes' force?
In general, here we can agree that the concept of weight makes sense only in empty space, where there is only one support and a body. What to do here, this is a question of terminology, which, unfortunately, everyone here has their own, since this is not such an important question :) And if the force of Archimedes in the air in all ordinary cases can be neglected, which means it has a special influence on the amount of weight cannot, then for a body in a liquid this is already critical.
To be completely honest, the division of forces into types is very arbitrary. Let's imagine a box being dragged along a horizontal surface. It is usually said that there are two forces acting on the box from the surface: a support reaction force directed vertically and a frictional force directed horizontally. But these are two forces acting between the same bodies, why don’t we simply draw one force, which is their vector sum (this, by the way, is sometimes done). Here, it's probably a matter of convenience :)
So I'm a little confused on what to do with this particular task. The easiest way is probably to reformulate it and ask a question about the magnitude of gravity.
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Statics is the branch of mechanics that studies the conditions of equilibrium of bodies.
From Newton's second law it follows that if the geometric sum of all external forces applied to a body is equal to zero, then the body is at rest or undergoes uniform linear 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 mass .
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.
In 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 in 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 relative to some axis, then for its equilibrium It is not enough for the resultant of all forces to be zero.
The rotating effect 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.
Product of the modulus of force per arm d called moment of force M. The moments of those forces that tend to turn the body counterclockwise are considered positive (Fig. 1.14.2).
Rule of Moments : a body having a fixed axis of rotation is in equilibrium if the algebraic sum of the moments of all forces applied to the body relative to this axis is equal to zero:
In the International System of Units (SI), moments of forces are measured in NNewton— meters (N∙m) .
In the general case, when a body can move translationally and rotate, for equilibrium it is necessary to satisfy both conditions: the resultant force being equal to zero and the sum of all moments of forces being equal to zero.
A wheel rolling on a horizontal surface - an example indifferent equilibrium(Fig. 1.14.3). If the wheel is stopped at any point, it will be in equilibrium. Along with indifferent equilibrium in mechanics, there are states sustainable And unstable balance.
A state of equilibrium is called stable if, with small deviations of the body from this state, forces or torques arise that tend to return the body to an equilibrium state.
With a small deviation of the body from a state of unstable equilibrium, forces or moments of force 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 protrusion is an example of unstable equilibrium. Finally, the ball at the bottom of the spherical recess 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. Indifference 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 straight line passing through the axis of rotation. Moreover, if the center of mass is below the axis of rotation, the state of equilibrium turns out to be stable. If the center of mass is located above the axis, the state of equilibrium is unstable (Fig. 1.14.5).
A special case is the balance of a body on a support. In this case, the elastic support force 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 support area, i.e. inside the contour formed by lines connecting the support points. If this line does not intersect the area of support, then the body tips over. An interesting example of the balance 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 is deviated 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 broken and the tower will fall when the deviation of its top from the vertical reaches 14 m. Apparently, this will not happen very soon.
