The vector sum of the developing forces. Law the folding of forces at the mechanics. How to know the sum and difference of vectors in coordinates

Mekhanichna diya tіl one on one zavzhda є їhnoy vzaєmodієyu.

If the body is 1, the body is 2, then at the same time the language is body 2, the body is 1.

For example,on the wire wheels of an electric locomotive (Fig. 2.3), the rails are pulled out of the side, rubbing calmly, straightening at the side of the wheel of the electric locomotive. The sum of these forces is the traction force of the electric locomotive. At their own line, the wheels are driven on the rails by the forces of rubbing calmly, straightening into the opposite wheel..

An ilkish description of the mechanical interplay was given by Newton in yoga the third law of dynamics.

For material points, this law formulated So:

Two material points work one on one with forces that are equal in magnitude and straightening in a straight line that moves the points(Fig.2.4):
.

The third law is not fair.

win suvoro

at different contact interactions,

at the same time it is quiet, that one is on the singing vіdstani, one is quiet, one is to rest.

Let's move on from the dynamics of the near material point to the dynamics mechanical systems, what is added up material points.

For - Three material points of the system, according to another Newton's law (2.5), maybe:

. (2.6)

Here і - masa that swidkіst -those material points, - the sum of all the forces that are on it.

The forces that work on a mechanical system are divided on the outside and inside. Evil forces blow on the points of the mechanical system from the side of other, outer bodies.

Internal forces run between points of the system itself.

Bring strength in viraz (2.6) it is possible to apply for seeing the sum of external and internal forces:

, (2.7)

de
– as a result of all the external forces that are on -that point of the system; - internal strength, sho drive on the qiu point from the side th.

Let us imagine that (2.7) y (2.6):

, (2.8)

having summed up the left and right parts of equals (2.8), recording for all material points of the system

. (2.9)

Behind Newton's third law, forces are mutual -Toy i -th point of the system is equal behind the module and protilege behind the direct
.

Therefore, the sum of all internal forces in equal (2.9) is equal to zero:

. (2.10)

The vector sum of all the forces that act on the system is called the main vector of the outer forces

. (2.11)

By changing the scope of the operation (2.9) by the scope of the operation, the difference and the reverse results (2.10) and (2.11)

- the main equalization of the dynamics of the progressive motion of a solid body

Tsіvnyannya vyslovlyuє the law of changing the impulse of a mechanical system: the time of the moment when the impulse of the mechanical system is equal to the head vector of the external forces that blow on the system.

2.6. Center ma ta law yogo ruhu.

mas center(inertia) mechanical system is called mottled , the radius-vector of which is the best way to create the mass of all material points of the system on their radius-vector to the mass of the entire system:

(2.12)

de і - mass and radius vector -those material points, -zagalna kіlkіst tsikh dot,
–total mass of the system.

How are the radius vectors drawn from the center of the mass , then
.

in such a manner, center mass - geometric point , for some sum of creations of the mass of all material points that establish a mechanical system, on the given radius-vector, drawn from the center of the point, to zero.

In the case of a non-stop distribution of the mass in the system (in the case of a lingering body), the radius-vector to the center of the mass of the system:

,

de ris the radius vector of a small element of the system, the mass of which is more expensivedm, integration is carried out for all elements of the system, tobto. by all weight m.

Having differentiated the formula (2.12) by the hour, we take

viraz for speed to the center of the mass:

Shvidkist center mas The mechanical system is more advanced in relation to the momentum of the system to the mass.

Todi system momentumdobutku її masi on swidk_st to the center of mass:

.

Substituting the vislav in the main equal dynamics of the progressive motion of a solid body, perhaps:

(2.13)

- the center of the mass of the mechanical system is collapsing like a material point, the mass is like a healthy mass of all systems and like a force that is equal to the head vector applied to the system of external forces.

Equation (2.13) shows that it is necessary to change the density of the mass center of the system, so that the system has an external force. The internal forces of the interplay of the elements of the system can cause changes in the speed of these elements, but they cannot be added to the total impulse of the system and the strength of the center of the mass.

If the mechanical system is closed, then
and the speed of the center of the mass does not change from hour to hour.

in such a manner, center of mass of a closed system or rest, or collapse with the constant speed of any inertial system in the wake. Tse means that the system can be tied to the center of the mass, and the system will be inertial.

With an one-hour surge, on one body, a lot of strength, the body collapses with quickening, which is quicker with a vector sum, like a winkle, under the influence of skin strength, okay. The force that is applied to the body is applied to one point, they are added according to the rule of folding vectors.

The vector sum of all forces, which simultaneously strikes the body, is called equal force and is determined by the rule of vector folding of forces: $\overrightarrow(R)=(\overrightarrow(F))_1+(\overrightarrow(F))_2+(\overrightarrow(F) ) _3+\dots +(\overrightarrow(F))_n=\sum^n_(i=1)((\overrightarrow(F))_i)$.

Equivalent strength can be on the body the same way, as the sum of all forces, as it is reported before.

For the folding of two forces, the parallelogram rule is used (Fig. 1):

Malyunok 1. Adding two forces behind the parallelogram rule

For which the modulus of the sum of two forces is known by the cosine theorem:

\[\left|\overrightarrow(R)\right|=\sqrt((\left|(\overrightarrow(F))_1\right|)^2+(\left|(\overrightarrow(F))_2\right |)^2+2(\left|(\overrightarrow(F))_1\right|)^2(\left|(\overrightarrow(F))_2\right|)^2(cos \alpha \ ))\ ]

If it is necessary to add up more than two forces applied at one point, then they are rooted by the rule of the bagatokutnik: from the end of the first force, a vector is drawn that is equal and parallel to the other force; from the end of another force - a vector, equal and parallel to the third force, then.

Figure 2

A flickering vector, drawing from the point of reporting forces to the end of the remaining force, for the magnitude of that directly more equal. In Fig. 2, this rule is illustrated by the application of the value of equal forces ~~of the forces $(\overrightarrow(F))_1,\ (\overrightarrow(F))_2,(\overrightarrow(F))_3,(\overrightarrow(F) )_4 $. It is important that under what vectors that are added up, it is not necessarily the fault of lying on one plane.

The result of di forces on a material point is to deposit only in the form of a module that directly. Firmly, the body can sing songs of remembrance. To that, however, behind the module and directly, the forces call out different fluctuations of the solid body fallowly from the point of stagnation. The straight line that passes through the force vector is called the line of force.

Baby 3

If the force is applied to different points of the body and not parallel to one, then it is equally applied to the point of crossing the line of force (Fig. 3).

The point rebuys at the level, as the vector sum of all forces that act on it, until zero: $\sum^n_(i=1)((\overrightarrow(F))_i)=\overrightarrow(0)$. For each direction, the sum of the projections of these forces onto all coordinates is equal to zero.

I will replace one force of two, applied at the same point and such that they vibrate the same force on the body, as if it were one force, call the distribution of forces. The distribution of forces vibrates, like that yoga addition, according to the rule of the parallelogram.

The task of laying out one force (the module and directly as it is in the house) on two, applied at one point and at the same time under the hood one to one, may be unambiguously resolved in such situations, as in the case of:

1. directly both storage forces;
2. the module is directly one of the warehouse forces;
3. modules of both storage forces.

Let's say, for example, we want to spread the force $F$ on two warehouses that lie in the same plane F and the direction of the straight lines a and b (Fig. 4). For which sufficient length of the vector, which represents F, draw two straight lines parallel to a and b. Vіdrіzki $F_A$ и $F_B$ represent the power consumption.

Figure 4. Deployment of the force vector behind the lines

The second variant of this task is to assign one of the projections of the force vector behind the given force vectors and another projection. (Fig.5 a).

Figure 5. The value of the projection of the force vector behind the given vectors

The task is to build up to a parallelogram along the diagonal and on one side, facing the planometry. In Fig.5b, such a parallelogram і is shown with the warehouse $(\overrightarrow(F))_2$ or $(\overrightarrow(F))$.

Another way to develop: increase the strength to the strength that increases $(\overrightarrow(F))_1$ (Fig. 5c). As a result, the force $(\overrightarrow(F))_2$ is taken away.

Three forces~$(\overrightarrow(F))_1=1\ N;;\ (\overrightarrow(F))_2=2\ N;;\ (\overrightarrow(F))_3=3\ N$ applied up to one point, lie in the same plane (Fig.6 a) and fold the cutie from the horizontal $\alpha =0()^\circ ;;\beta =60()^\circ ;;\gamma =30()^\ circ $ obviously. Know these equal forces.

We draw two mutually perpendicular axes OX and OY so that all OX bends along the horizontal, which directs the force $(\overrightarrow(F))_1$. p align="justify"> We design the force data on the coordinate axis (Fig. 6 b). The projections $F_(2y)$ and $F_(2x)$ are negative. The sum of the projections of forces on all of the good projections on the qi of all the equals: $F_1+F_2(cos \beta \ )-F_3(cos \gamma \ )=F_x=\frac(4-3\sqrt(3))(2)\ approx -0.6\H$. Similarly, for projections onto the whole OY: $-F_2(sin \beta \ )+F_3(sin \gamma =F_y=\ )\frac(3-2\sqrt(3))(2)\approx -0.2\ H$ . The equal modulus is determined by the Pythagorean theorem: $ F = \ sqrt (F ^ 2_x + F ^ 2_y) = \ sqrt (0.36 + 0.04) \ approx 0.64 \ H $. Directly, it is equally significant for the additional kuta between equal and fluent (Fig. 6c): $ tg \ varphi = \ frac (F_y) (F_x) = \ frac (3-2 \ sqrt (3)) (4-3 \ sqrt ( 3))\approx 0.4$

The force $F = 1kH$ is applied at point B of the bracket i and is straightened vertically to the bottom (Fig. 7a). Find out the warehouse price of the force for the straightening of the shear bracket. Required data to indicate the little one.

F = 1 kN = 1000N

$(\mathbf \beta )$ = $30^(\circ)$

$(\overrightarrow(F))_1,\ (\overrightarrow(F))_2$ - ?

Let the haircuts be attached to the wall at points A and C. The distribution of the force $(\overrightarrow(F))$ in the warehouses straight AB and BC is shown in Fig. 7b. The stars show that $ \ left | (\overrightarrow(F))_1\right| = Ftg\beta\approx 577\H; \ \ $

\[\left|(\overrightarrow(F))_2\right|=F(cos \beta \ )\approx 1155\ H. \]

Response: $ \ left | (\overrightarrow(F))_1\right| $ = 577 N; $\left|(\overrightarrow(F))_2\right|=1155\ H$

How do you see the folding of vectors, how do you know how to learn. Children do not show what is after them. To be brought simply to remember the rules, and not to think about the essence. To that very thing about the principles of folding and understanding of vector quantities, it is necessary to know a lot.

The result of adding two and more vectors will always have one more. Moreover, the wine will be bound by obov'yazkovo will be the same regardless of the acceptance of this sign.

Most of all, in the school course in geometry, the folding of two vectors is seen. Vono can buti vikonane for the rule of trikutnik chi parallelogram. Qi little ones look different, but the result is the same.

How do you follow the rule of trikutnik?

It will stop only if the vectors are not collinear. To not lie on one straight line or parallel ones.

At this point, it is necessary to include the first vector in the direction of the first point. For the third time, it is necessary to draw parallel and equal to another. The result will be a vector that starts from the cob of the first and ends like the other. The little one is guessing a trickster. Zvіdsi th name of the rule.

Although vectors are collinear, the same rule can be zastosovuvati. Less than little ones, there will be roztashovaniya vzdovzh one line.

How do you follow the parallelogram rule?

I know? zastosovuetsya less for non-collinear vectors. Pobudova vykonuєtsya for іnshim principle. I want an ear like this. It is necessary to include the first vector. The first type of yoga on the cob is different. On their basis, obtain a parallelogram and draw a diagonal from the cob of both vectors. Won be the result. This is how the folding of vectors follows the parallelogram rule.

Dosi їх it was two. And yak buti, yakscho їх 3 chi 10? Vikoristovuvati offensive reception.

How and if the rule of the bagatokutnik is enforced?

As a rule, it is necessary to fold the vectors, the number of which is more than two, it is not varto. Sufficiently consequently, it is necessary to add їх mustache and z'єdnati ear of the lancelet with її kinets. Tsey vector i will be a shukano bag.

What is the authority of the authorities for the diy of the vectors?

About the zero vector. Yake stverzhuє, scho when you add him to go out.

About the prolongation vector. Tobto about such a thing that can be straight forward and equal to the value module. Їх sum is equal to zero.

About commutative folding. Those who are seen post school. Change the scope of dodankiv not to lead to a change in the result. In other words, whether there is a vector to work backwards. Let everything be true and united.

About the associativity of folding. Whose law allows you to put together in pairs whether vectors from three and add a third to them. If you write it down for additional signs, then you will see this:

first + (other + third) = other + (first + third) = third + (first + other).

What do you know about the difference in vectors?

There is no sound operation. Tse z tim, sho yogo, vlasne, є dodavannyam. Only another one of them is given a straight line. And then we all win in such a way that we could not see the folding of the vectors. It’s practical not to talk about their retail.

In order to ask the robot to see it, the rule of trikutnik has been modified. Now (with vіdnimanni) another vector needs to be added from the cob of the first. Vidpoviddu will be those who see the last point of the change with her. If you want to, you can do it the way it was described earlier, just changing the other directly.

How to know the sum and difference of vectors in coordinates?

The task has given the coordinates of the vectors and it is necessary to recognize their values ​​for the sub-bag. Under what circumstances it is not necessary to win. So you can use clumsy formulas to describe the rule of folding vectors. The stinks look like this:

a(x, y, z) + (k, l, m) = s(x + k, y + l, z + m);

a (x, y, z) - (k, l, m) = s (x-k, y-l, z-m).

It is easy to remember that the coordinates need to be simply added together to see the fallow in a particular plant.

The first example of solutions

Wash away. Given a rectangle ABCD. The sides are 6 and 8 cm long.

Solution. On the back of the page, display q vectors. The stench is straightened from the vertices of the rectangle to the point of the crossbar of the diagonals.

If you respectfully marvel at the armchair, then you can tell that the vectors have already been taken so that the other of them sticks with the end of the first. The axis of only yoga is directly wrong. Vіn maє z tsієї points start. All the vectors are added up, and the tasks are visible. Stop. Tsya diya means that it is necessary to add an opposite vector. So, it is necessary to replace VO with BB. I see that two vectors have already made a couple of sides out of the tricot rule. That is why the result of their own folding, tobto difference, to joke, is the vector AB.

And the wine is zbіgaєtsya from the side of the rectangle. In order to write down the numerical value, you need to take this. Cross the rectangle in such a way that the larger side goes horizontally. Begin the numbering of the vertices from the lower left and go on the opposite arrow. Then the length of the vector AB is 8 cm.

Vidpovid. R_nitsya AT i VO dorіvnyuє 8 cm.

Another butt of that yoga report is not a solution

Wash away. The rhombus ABSD has a diagonal line of 12 and 16 cm.

Solution. Let the recognition of the vertices of the rhombus be the same as in the front task. It is similar to the decision of the first butt to enter, what is different, what is joking, closer to the vector AB. And yoga is not at home. The solution of the problem was carried out before the calculation of one of the sides of the rhombus.

For whom it is necessary to look at the ABO tricoutnik. Vin is straight-cut, to that the diagonals of the rhombus are tinted at the apex of 90 degrees. And the yoga legs are lined with half of the diagonals. So 6 and 8 cm.

For її znakhodzhennya the Pythagorean theorem is necessary. The square of the hypotenuse is equal to the sum of the numbers 6 2 і 8 2 . After the zvedennya at the square, the values ​​\u200b\u200bare seen: 36 and 64. Їx sum - 100. It is clear that the hypotenuse is 10 div.

Vidpovid. The difference in the vectors of AT and BO is 10 cm.

The third example from detailed solutions

Wash away. Calculate the difference and the sum of two vectors. Vіdomi їх coordinates: for the first - 1 and 2, for the other - 4 and 8.

Solution. To know the sum, it is necessary to put together the first and other coordinates in pairs. The result will be the numbers 5 and 10. It will be a vector with coordinates (5; 10).

For retail, it is necessary to have a visual representation of the coordinates. After vikonanny tsієї dії see the numbers -3 and -6. The stink will be the coordinates of the shuk vector.

Vidpovid. The sum of the vectors - (5; 10), their difference - (-3; -6).

Quarter butt

Wash away. The length of the vector AB is 6 cm, ND - 8 cm. Calculate: a) the difference between the vector modules VA and PS and the module of the difference between VA and PS; b) the sum of the modules and the module of the sum.

Solution: a) Longer vectors have already been given a task. Therefore, calculate their cost in the warehouse of hardships. 6 - 8 = -2. The situation with the retail module is somewhat more complicated. On the back, it is necessary to recognize, which vector will be the result of investigation. Z tsієyu by the method of following the vector BA, which is the directing of the protilege AB. Let's draw the BC vector from the beginning, directing it to the BC, opposite to the VC. The result of visualization is the CA vector. Its module can be determined by the Pythagorean theorem. Just calculate to the value of 10 cm.

b) The sum of the modules of the vectors is equal to 14 cm. Vector BA protilezhno straightening up to what is given - AB. Offending vectors and directing from one point. In this situation it is possible to overcome the rule of the parallelogram. The result of adding will be a diagonal, and not just a parallelogram, but a rectangle. Yogo diagonally equal, later, the sumi module is the same, like at the front point.

Vidpovid: a) -2 and 10 cm; b) 14 and 10 cm.

Section 1. "STATICS"

Newtony

The shoulder of the force is the shortest distance from the point to the line of force

Tvіr forces on the shoulders dovnyuє moment of strength.

8. Formulate a “rule right hand» for the purpose of directing the moment of force.

9. How is the head moment of the system of forces determined at a point?

The head moment to the center is the vector sum of the moments of all forces applied to the body to the same center.

10. What is called a pair of forces? Why is the moment of the betting of forces worth? Chi lie down in vіd vіd vyboru points? How to direct and why do you care about the magnitude of the moment of the bet of forces?

A pair of forces is called a system of forces among equal forces, parallel and parallel to one another. The moment to dobutka one of the forces on the shoulders, do not lay down in the choice of a point, straightening perpendicularly to the plane in a pair.

11. Formulate the Poinsot theorem.

If a system of forces, which is absolutely solid, can be replaced by one force and one pair of forces. For whom, the force will be the head vector, and the moment of the bet will be the head moment of the entire system of forces.

12. Formulate the necessary enough mind equal systems of forces.

For a flat system of forces to be equal, it is necessary and sufficient, so that the sum of the algebra of projections of all forces on two coordinate axes and the algebraic sum of the moments of all forces in a visually sufficient point is equal to zero. Another form of equality of equality is the equality to zero of the algebraic sums of the moments in the forces of all three points, so that they do not lie on one straight line

14. What systems of forces are called equivalent?

However, without destroying the body, one system of forces (F 1 , F 2 , ..., F n) can be replaced by another system (P 1 , P 2 , ... , P n) and at the same time, such systems of forces are called equivalent

15. What force is called equal to the given system of forces?

If the system of forces (F1, F2, ..., Fn) is equivalent to one force R, then R ranks. equal. An equal force can replace the effect of all these forces. Ale, don’t be like a system of forces can be equal.

16. It seems that the sum of the projections of the forces applied to the body, on the whole is equal to zero. How direct is such a system?

17. Formulate the axiom of inertia (the principle of inertia of Galileo).

Under the influence of forces, which are vrіvnovazhuyus, the material point (body) is in the camp of calm, but it collapses straight and evenly.

28. Formulate the axiom of the equality of two forces.

Two forces, applied to an absolutely solid body, will be equally important either, if the stink is equal behind the module, blowing on one straight line and on the opposite side

19. Is it possible to endure the force of the air of a line without changing the kinematic state of an absolutely solid body?

Without changing the kinematic state of an absolutely solid body, the force can be transferred to the vzdovzh linії її diї, taking the invariable її modulus directly.

20. Formulate the axiom of the parallelogram of forces.

Without changing the body, two forces, applied to one point, I can replace one equal force, applied at the same point and equal to their geometric sum

21. How is Newton's third law formulated?

Be-yakіy dії vіdpovіdaє rіvna i protilezhno pryatvovana protidia

22. What is a firm body called wrong?

The forces that exist between the bodies of the system are called internal.

Hinged-rolling support. This type of linkage is structurally shaped like a cylindrical hinge, which can move freely on the surface. The reaction of the hinged-rolling support of the plant is straightened perpendicularly to the support surface.

Hinged-unbreakable support. The reaction of a hinged-non-destructive support is given at the sight of non-domic storage and lines of parallel lines or they are aligned with the coordinate axes

29. What kind of support is called a hard mortgage (tack)?

This is an unusual view of the link, so like a crim moving in the plane, a horst of larynx is crossing the turn of the shear (beam) at a point. Therefore, the reaction is connected as before the reaction ( , ), before the reactive moment

30. What kind of support is called a pilgrimage?

A support and a spherical hinge This kind of tie can be applied to a haircut, which may be on the end of a spherical surface, as to be fixed in a support, which is a part of a spherical empty. The spherical hinge is shifting along a straight line in space, so the reaction of yoga is represented by looking at three warehouses, parallel to the coordinate axes

31. What support is called a spherical hinge?

32. What system of forces is called similar? How are the minds of equal systems of forces formulated, how to converge?

Yakshcho (absolutely firm) the body is found in equal parts of a flat system of three non-parallel forces (these forces, for which two non-parallel forces want to be), the lines of these two are intertwined in one point.

34. Why is the sum of two parallel forces straightened into one bead worth? Have different sides?

equal to two parallel forces F 1 and F 2 of one directly can be the same directly, її the module is additional to the sum of the modules of additional forces, and the point of addition to divide between the points of the report of forces on the part is wrapped in proportion to the modules of forces: R \u003d F 1 + F 2; AC/BC=F2/F1. Equally two parallel parallel forces can directly force more behind the module and the module, which is more expensive than the difference between force modules.

37. How is Varignon's theorem formulated?

If the plane of the system of forces, which is seen, is reduced to equal, then the moment of the equal equal to the point is equal to the algebraic sum of the moments of the given system’s forces, the moment is equal to the eye of the point itself.

40. How is the center of parallel forces defined?

Behind Varignon's theorem

41. How is the center of the body of the hard body defined?

45. Where is the center of gravity of trikutnik?

Crosspoint median

46. ​​Where is the center of the pyramid and cone?

Section 2. "KINEMATICS"

1. What is the point trajectory called? Which direction of the point is called rectilinear? Curvilinear?

Line, vzdovzh as the material crumbles mottled , call a trajectory .

If the trajectory is a straight line, then the point is called a straight line; if the trajectory is a crooked line, then the movement is called curvilinear

2. How is the Cartesian rectangular coordinate system defined?

3. How is the absolute stability of a point determined in a non-rigid (inertial) coordinate system? How to straighten the vector of stability how її trajectory? What is the purpose of projecting the width of a point on the axes of Cartesian coordinates?

For the point qi of fallowness, these are:

.

3. How are absolutely accelerated points determined in a non-robust (inertial) coordinate system? Why do projections of accelerated points on the axis of Cartesian coordinates?

5. How does the vector of the apex stiffness of a solid body appear when it is wrapped around a slightly unbreakable axis? How to straighten the vector of the top joint?

Kutova swidkіst- Vector physical quantity, which characterizes the firmness of body wrapping. The vector of the corner of the wind speed by the value of the curve is equal to the turn of the body in one hour:

and straightening along the axis, the wrapping was twisted with the rule, so, in that bek, in some bi, twisting the sverdlik with the right cuts, yakby wrapping in the same direction.

6. How does the vector of the apex acceleration of a solid body appear when it is wrapped around a slightly unbreakable axis? How is the vector of the apex straightened?

With the wrapping of the body on a slightly indestructible axis, the kutov is accelerated modulo dorіvnyuє:

The vector of the apex acceleration α straightening of the uzdovzh osі wrapping (killed with an accelerated wrapping and protilezhno - with an uplifted one).

When wrapped around a slightly non-destructive point, the vector of the apex speed is shown as the first one is similar to the vector of the apex of the speed per hour, then

8. Why is equal absolute, figurative, and transparent to the point at її foldable Russian?

9. How are the transference and speed of the moment when the Russian point is collapsible?

10. How does the speed of the coriole signify with a folding Russian dot?

11. Formulate the Corioles theorem.

Quick folding theorem (Corioles theorem): , de – Coriole acceleration (Coriolis acceleration) – in the case of a non-transferable portable rush, absolutely acceleration = geometric sum of a portable, visual and Coriolis acceleration.

12. Under some conditions, the points equal to zero:

a) how fast?

b) is it normal to hurry up?

14. What movement of the body is called progressive? Why is it equal to speed and quickening of the point of the body in such a Russian?

16. What kind of body movement is called wraparound? Why is it equal to speed and quickening of the point of the body in such a Russian?

17. How do those pre-centre accelerating points of a solid body turn more precisely, which wrap around on a nearly unbreakable axis?

18. What is a geometrically spaced point of a solid body that wraps around an unbreakable axis, which at a given moment may have the same value and the same direction?

19. What kind of movement of the body is called plane-parallel? Why is it equal to speed and quickening of the point of the body in such a Russian?

20. How does the mittevian center of the shvidkos of a flat figure appear, which collapses at its own plane?

21. How can you graphically know the position of the mitt to the center of the pins, as if you can see the pins of two points of a flat figure?

22. What will be the sharpness of the point of the flat figure at the tip, if the center of the wrap around the figure is not cut off?

23. How do you tie together the projections of the flatness of two points of a flat figure on a straight line, how do you connect the points?

24. Given two points ( BUTі At) flat figure that collapses, moreover, it seems that the point is swept BUT perpendicular to AB. How the straightness of the point is straightened At?

Section 1. "STATICS"

1. What factors determine the strength that is on the firm

2. In some individuals, the strength of the "CI" system is victorious?

Newtony

3. Why is the head vector of the system of forces important? How to induce a forceful bagatokutnik for a given system of forces?

The head vector is the vector sum of all forces that reach the body

5. What is called the moment of force for each point? How to straighten out the moment of force, how about the vector of force and the radius vector of the point of the report of the force?
The moment of force at a point (center) is a vector that is numerically equal to the additional force module on the shoulder, so that it is the shortest distance from the designated point to the line of force. Vіn straightening perpendicular to the plane of the expansion of the force and r.v. points.

6. How many times the moment of force should the points reach zero?
If the shoulder is good 0

7. How is the shoulder strength or shodo point determined? Why do you need more strength on your shoulder?

When you pour in a lot of strength on one body, the body suddenly begins to crumble with quickening, which is quicker with a vector sum, like a winickle under a surge of skin strength, okay. Up to the forces that work on the body, adding to one point, the rule of folding vectors is established.

Appointment 1

The vector is the sum of all forces, which is poured into the body at once, the whole force equal, Yaka depends on the rule of vector folding of forces:

R → = F 1 → + F 2 → + F 3 → +. . . + F n → = ∑ i = 1 n F i → .

Equivalent force strikes the body as it is, like the sum of all forces that strike the new.

Appointment 2

For folding 2 vicorist forces rule parallelogram(baby 1).

Malyunok 1 . Adding 2 forces behind the parallelogram rule

Let us derive the formula for the modulus of equal force using the additional cosine theorem:

R → = F 1 → 2 + F 2 → 2 + 2 F 1 → 2 F 2 → 2 cos α

Appointment 3

If necessary, add more than 2 vicorist forces bagatokutnik's rule: type of kintsia
1st force it is necessary to draw a vector equal and parallel to the 2nd force; in the end of the 2nd force, it is necessary to draw a vector equal and parallel to the 3rd force, etc.

Malyunok 2 . The folding of forces by the rule of the bagatokutnik

Ending vector, drawing from the point of reporting forces at the end of the remaining force, for the magnitude of that direct additional equal force. Baby 2 at first glance illustrates the butt of knowing equal forces from 4 forces: F 1 → , F 2 → , F 3 → , F 4 → . Moreover, the vectors, which are supposed to be summed up, are, of course, neobov'yazkovo guilty but in the same plane.

The result of the force on the material point is less than in the form of a module that directly. Solid body - songs of rosemary. That is why the forces with the same modules and directly call out different fluctuations of the solid body fallowly from the point of stagnation.

Appointment 4

Line of diy forces name a straight line that passes through the force vector.

Malyunok 3 . Folding forces, adding to the life points of the body

Likewise, forces are applied to different points of the body and not parallel to one to one, but equally applied to the point of crossing the line of force (figure 3 ). The point is rebuffed in equal position, as the vector sum of all forces, which act on it, is equal to 0: ∑ i = 1 n F i → = 0 → . At to this particular type dorіvnyuє 0 that sum of projections of these forces on the coordinate everything.

Appointment 5

Distribution of forces in two warehouses- for the replacement of one force of two, applied at the same point and such as to crush the same body on the body, as it is one force. The distribution of forces is established, as if adding, by the rule of the parallelogram.

The task of laying out one force (of a module and directly as a task) for 2, applied at one point and one at a time, one to one, may be unambiguously resolved in such situations, if in a house:

• straight 2 warehouse forces;
• the module is directly one of the warehouse forces;
• modules of 2 storage forces.

butt 1

It is necessary to distribute the force F on 2 warehouses, which are located in the same plane from F and the direction of the airflow straight lines a and b (figure 4 ). Then it is sufficient to draw 2 straight lines parallel to straight lines a and b. Vіdrіzok F A and vіdrіzok F B represent shukanі force.

Malyunok 4 . Deployment of the force vector behind the lines

butt 2

Another variant of this task is to know one of the projections of the force vector behind the given force vectors and 2 projections (Figure 5 a).

Malyunok 5 . The value of the projection of the force vector behind the given vectors

In another option, it is necessary to create a parallelogram along the diagonal and on one side, like in planometry. On the little 5b, such a parallelogram is depicted and assigned to the shukan warehouse F 2 → or F →.

Also, the 2nd way of rozv'azannya: increase the strength to strength, which is good - F 1 → (Figure 5 c). The result will require a force F → .

butt 3

Three forces F 1 → = 1 N; F 2 → = 2 N; F 3 → = 3 β = 60°; γ = 30° It is necessary to know equal strength.

Solution

Malyunok 6 . Znahodzhennya equal force behind given vectors

At least mutually perpendicular to the axis O X and O Y in such a way that all O X swayed from the horizontal, which directs the force F 1 →. Let's look at the projection of these forces on the coordinate axis (Figure 6 b). Projections F 2 y and F 2 x are negative. The sum of the projections of forces on the coordinate axis O X is a good projection on the qiu of the balance: F 1 + F 2 cos β - F 3 cos γ \u003d F x \u003d 4 - 3 3 2 ≈ - 0.6 N.

So it is for projections on the whole O Y: - F 2 sin β + F 3 sin γ \u003d F y \u003d 3 - 2 3 2 ≈ - 0, 2 N.

The modulus of equal variation is significant behind the additional Pythagorean theorem:

F \u003d F x 2 + F y 2 \u003d 0.36 + 0.04 ≈ 0.64 N.

We directly know for the help of the kuta between the equal weight and the weight (Figure 6 c):

t g φ \u003d F y F x \u003d 3 - 2 3 4 - 3 3 ≈ 0.4.

butt 4

Force F = 1 kN is applied at point B of the bracket i straightened vertically to the bottom (Figure 7 a). It is necessary to know the warehouse strength for the straight shear bracket. Mustache necessary data for the little one

Solution

Malyunok 7 . The value of the storage force F behind the straightening of the shear bracket

Given:

F = 1 to H = 1000 N

Let the haircuts be screwed to the wall at points A and C. On the little 7 b, the distribution of force F is depicted → on the warehouse vzdovzh directly AB and B C.

F 1 → = F t g β ≈ 577 N;

F 2 → = F cos β ≈ 1155 N.

Suggestion: F 1 → = 557 N; F 2 → = 1155 n.

How did you remember the pardon in the text, be kind, see it and press Ctrl + Enter