Newton's first law tells us that in inertial frames of reference, bodies can change speed only if they are influenced by other bodies. With the help of force ($\overline(F)$) they express the mutual action of bodies on each other. A force can change the magnitude and direction of a body's velocity. $\overline(F)$ is a vector quantity, that is, it has a modulus (magnitude) and direction.
Definition and formula of the resultant of all forces
In classical dynamics, the main law by which the direction and magnitude of the resultant force is found is Newton’s second law:
\[\overline(F)=m\overline(a)\ \left(1\right),\]
where $m$ is the mass of the body on which the force $\overline(F)$ acts; $\overline(a)$ is the acceleration that the force $\overline(F)$ imparts to the body in question. The meaning of Newton's second law is that the forces that act on a body determine the change in the speed of the body, and not just its speed. You should know that Newton's second law is true for inertial frames of reference.
Not one, but a certain combination of forces can act on a body. The total action of these forces is characterized using the concept of resultant force. Let several forces act on a body at the same moment in time. The acceleration of the body in this case is equal to the sum of the acceleration vectors that would arise in the presence of each force separately. The forces that act on the body should be summed up in accordance with the rule of vector addition. The resultant force ($\overline(F)$) is the vector sum of all forces that act on the body at the considered moment in time:
\[\overline(F)=(\overline(F))_1+(\overline(F))_2+\dots +(\overline(F))_N=\sum\limits^N_(i=1)((\ overline(F))_i)\ \left(2\right).\]
Formula (2) is the formula for the resultant of all forces applied to the body. The resultant force is an artificial quantity that is introduced for the convenience of calculations. The resultant force is directed as the acceleration vector of the body.
The basic law of the dynamics of translational motion in the presence of several forces
If several forces act on a body, then Newton's second law is written as:
\[\sum\limits^N_(i=1)((\overline(F))_i)=m\overline(a)\left(3\right).\]
$\overline(F)=0$, if the forces applied to the body cancel each other out. Then in the inertial reference frame the speed of the body is constant.
When depicting the forces acting on a body in the figure, in the case uniformly accelerated motion, the resultant force, is depicted as longer than the sum of the forces that are directed opposite to it. If the body moves at a constant speed or is at rest, the lengths of the force vectors (the resultant and the sum of the remaining forces) are the same and they are directed in opposite directions.
When the resultant of the forces is found, all the forces taken into account in the problem are shown in the figure. These forces are summed up in accordance with the rules of vector addition.
Examples of problems on resultant forces
Example 1
Exercise. A material point is acted upon by two forces directed at an angle $\alpha =60()^\circ $ to each other. What is the resultant of these forces if $F_1=20\ $N; $F_2=10\ $H?
Solution. Let's make a drawing.
Forces in Fig. We add 1 according to the parallelogram rule. The length of the resultant force $\overline(F)$ can be found using the cosine theorem:
Let's calculate the module of the resultant force:
Answer.$F=26.5$ N
Example 2
Exercise. Forces act on a material point (Fig. 2). What is the resultant of these forces?
Solution. The resultant of the forces applied to the point (Fig. 2) is equal to:
\[\overline(F)=(\overline(F))_1+(\overline(F))_2+(\overline(F))_3+(\overline(F))_4\left(2.1\right).\]
Let us find the resultant of the forces $(\overline(F))_1$ and $(\overline(F))_2$. These forces are directed along the same straight line, but in opposite directions, therefore:
Since $F_1>F_2$, then the force $(\overline(F))_(12)$ is directed in the same direction as the force $(\overline(F))_1$.
Let us find the resultant of the forces $(\overline(F))_3$ and $(\overline(F))_4$. These forces are directed along one vertical straight line (Fig. 1), which means:
The direction of the force $(\overline(F))_(34)$ coincides with the direction of the vector $(\overline(F))_3$, since $(\overline(F))_3>(\overline(F))_4 $.
We find the resultant that acts on the material point as:
\[\overline(F)=(\overline(F))_(12)+(\overline(F))_(34)\left(2.2\right).\]
The forces $(\overline(F))_(12)$ and $(\overline(F))_(34)$ are mutually perpendicular. Let's find the length of the vector $\overline(F)$ using the Pythagorean theorem:
This is the vector sum of 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 side of the earth give a resultant force that imparts centripetal acceleration required for circular motion
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 with uniform acceleration. 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, to correctly designate all the forces acting on the body; then draw coordinate axes, select their directions; in the third step it is necessary to determine the projections of the vectors on the axes; 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.
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.
