With a couple of forces
is a system of two equal in magnitude, parallel and directed in opposite directions forces acting on a solid body (Fig. 2.1.15). Plane 
, containing the action lines of the pair forces 
And 
called plane of action of the couple forces
. Distance 
between the lines of action of the pair forces is called couple's shoulder
.
Let us determine the value of the moment of the pair relative to an arbitrary center ABOUT. According to the parallelogram rule - couple moment vector equal to the geometric sum of the moments of the couple forces 
And 
, i.e.,
from where, taking into account that, by definition, a pair of forces 
, we get
The modulus of the moment of the pair is equal to
The algebraic value of the moment of a pair of forces is equal to
, Where 
.
(2.1.22)
The couple's moment counts positive , if he tends to rotate the body counterclockwise and negative , if - clockwise.
From expressions (2.1.20) and (2.1.21) it is clear that the vector-moment 
pairs perpendicular to the plane of action of forces 
And 
and does not depend on the position in space of the center ABOUT, since, wherever we choose the center, the vector 
retains its meaning. Thus, without disturbing the magnitude and direction of the momentum vector of the pair 
, plane of action 
The forces of the pair can be transferred in parallel in any way in space.
Based on the above, we can formulate the following properties of pairs. The action of a couple on a solid body will not change if:
1) move the pair in the plane of its action to any other position;
2) transfer the pair to any other plane parallel to the plane of its action;
3) the modulus of the couple’s forces is increased (or decreased) several times, and its leverage is decreased (or increased) by the same amount.
If pairs of forces are located in the same or parallel planes, then they are added algebraically. If pairs of forces are located in intersecting planes, then they add up geometrically.
IN 
In technical problems, it is necessary to take into account not only sliding friction, but also the so-called rolling friction, the measure of which is moment of rolling friction pair. Consider a cylindrical roller lying on a horizontal plane (2.1.16,a). If there are no active forces other than gravity 
, do not act on the skating rink, then the force 
balanced by normal reaction 
the supporting surface and the roller remains at rest.
Let us apply a horizontal force to the roller 
(Fig. 2.1.16, b). Then at the point
touch A friction force will arise 
, preventing the roller from sliding along the plane, and the point of application of the normal reaction 
will move relative to the point A in the direction of the force 
some distance h. This is explained by the fact that, due to deformation, the roller actually comes into contact with the plane along a small area centered at the point A. After application of force 
, the load on the left half of the site will decrease, and on the right half it will increase. As a result, the line of action of the normal reaction 
will shift to the right and a pair of forces will arise ( 
,
) with shoulder h and the moment 
. This pair, which prevents the roller from rolling along the plane, is called a rolling friction pair, and its moment 
- rolling friction moment.
With increasing strength 
from zero leverage h and moment 
increase to the limiting values at which the roller begins to roll:
. (2.1.23)
Magnitude 
, which has the dimension of length, is called the rolling friction coefficient; its values are determined experimentally and are given in technical reference books. Coefficient 
and moment 
the less, the harder the contacting bodies and the cleaner their surfaces are processed.
After the start of rolling, the moment 
regardless of the magnitude of the force 
equal to the limit value 
.
2.1.8. System of forces. Main vector and main moment of the force system System of forces called any set of forces acting on a body
or mechanical system at the same time. Any system of forces can be brought to an arbitrary center by replacing it with an equivalent system of forces. Consider bringing one force to a given center, not lying on the line of action of this force(Poinsot method).
Let to a free rigid body at the point 
force applied 
(Fig. 2.1.17). Let's take an arbitrary point 
(center of adduction) and draw the force through it 
plane 
. Let's put it in the center 
balanced system of forces 
,
; equal in modulus 
and parallel to it. Force system 
equivalent to force 
. On the other hand, it can be considered as consisting of force 
, geometrically equal to the force 
, but attached in the center 
, and couples 
, called annexed
. It is easy to see that the moment of the attached pair 
geometrically equal to the moment of force 
relative to the center 
:
. Using the symbol of equivalence of systems of forces, we write 
∞
.
So, a force applied at any point of the body is equivalent to an equal force applied at an arbitrarily chosen center, and a pair whose moment is equal to the moment of the given force relative to this center.
Let a system of forces act on a free rigid body 
, located arbitrarily in space and applied at points 
. Let us bring all these forces to an arbitrary center 
. As a result, we get strength 
, equal to the given forces and applied at the center 
and associated pairs. Moments 
of these attached pairs are equal to the moments of these forces relative to the center of reduction:
Adding up strength 
, attached in the center 
according to the polygon rule, we get one force 
. Since the strength 
equal to geometrically given forces 
, then we can write 
.
(2.1.25)
Vector 
, equal to the geometric sum of all forces of the system, is called main vector
systems of forces.
Adding the adjoint pairs, we get one pair with a moment 
, equal to the geometric sum of the moments of the associated pairs 
.
(2.1.26)
Taking into account (2.1.24), we find
.
(2.1.27)
Vector 
, equal to the geometric sum of the moments of all forces of the system relative to the center of reduction 
, called main point
systems of forces relative to this center.
Thus, an arbitrary system of forces applied to a free rigid body can be reduced to one force equal to the main vector of the force system, and applied at the center of reduction and to one pair with a moment equal to the main moment of this system relative to the center of reduction. The main vector should not be identified 
c resultant, since it replaces the system of forces in combination with the main moment, while the resultant 
, if it exists, one replaces the system of forces.
When the center of reduction is transferred, the main vector does not change, but the main moment in the general case changes.
Self-test questions on topic 2.1
Why is force a vector quantity?
What is the inertia of material bodies?
If a point does not interact with other material bodies, then in what state should it be?
Formulate the basic law of mechanics.
Expand the force vector along the coordinate axes.
What principle of mechanics allows us to study the mechanics of non-free material bodies?
In what case is the moment of a force about a point equal to zero?
What is the moment of force about an axis?
In what cases is the moment of a force about an axis equal to zero?
What is a force couple? What is the moment of the couple?
What is the direction and where is the moment vector of the couple applied?
Formulate the properties of a pair of forces.
How pairs lying in the same plane are formed; in intersecting planes?
Determine the moment of the rolling friction pair.
Apply the force to any arbitrary point of the rigid body.
16. What is called the main vector of a force system?
17. Define the main moment of an arbitrary system of forces relative to the center of reduction.
18. Will the main vector and the main moment of the system of forces change when the center of adduction is moved to another position?
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Pair
- J. German. a couple, a couple, a friend, a deuce, a deuce, a deuce, a deuce, a twin, a deuce; two objects taken together for some reason, at the same time.... and 1 more definition Dahl's Dictionary
Pair
- Silver coin of Turkey, issued in 1623. Silver coin of the Crimean Khanate in the 17th century. Copper coin issued by Russia in 177... Numismatist's Dictionary
Pair
- A small amount of something, a few. and 7 more definitions Ozhegov's Dictionary
Pair
- - piece - a silver coin of Turkey, and in circulation since 1623, since 1930 it has been used only as a counting unit equal to one forty... Historical Dictionary
Pair
- (Sanskrit) In philosophy - “limitless” and “supreme”, the final boundary. Param is the end and purpose of existence; Parapara is pr... Philosophical Dictionary
Pair
- Belem, Sao Francisco. Dictionary of geographical names
PAIR
- Y, plural. couples, couples, w. The coin of change of Turkey, equal to 1/40 of the kurus (piastre), and of Yugoslavia, equal to 1/100 of the dinar. and 1 more definition Dictionary of foreign words
pair
- Since 1696, Peter I; see Christiani 43; Ukrainian, blr. pair. Borrowing via Polish para from Middle-European-N. par "pair" from Lat. par "equal, par... Vasmer's Etymological Dictionary
pair
- P "ARA, couples, female (German Paar). 1. Two homogeneous objects, used together and forming one whole, a comp... Ushakov's Explanatory Dictionary
PAIR meaning
T.F. Efremova New dictionary of the Russian language. Explanatory and word-formative
Meaning:
P A ra
1. and.1) Two homogeneous or identical objects, used together and forming one whole or set.
3) Two as something whole.
a) colloquial Two beings suitable for each other.
b) One who suits another for some reason. qualities, in some way respect.
a) outdated A man and a woman approaching each other as spouses.
b) One who is suitable for another to be a wife.
a) Two horses harnessed to a carriage or cart.
b) Crew, a cart with two harnessed horses.
7) decomposition Two pieces of smth.
2. and. up-downAn assessment of a student's performance that means poor, unsatisfactory; deuce.
Modern explanatory dictionary ed. "Great Soviet Encyclopedia"
Meaning:
city in Brazil; see Belen.---(Turkish para, from Persian para - piece), 1) silver coin of Turkey, in circulation since 1623; since 1930 it has been used only as a unit of counting, equal to 1/40 kurus. 2) Silver coin of the Crimean Khanate (17th century). 3) Copper coin issued by Russia for Moldova and Wallachia during the period of hostilities in 1771-74.---... (from the Greek para - near, past, outside), 1) part of compound words, meaning: located nearby, as well as deviating from something, violating something (for example, paramagnetism). 2) In chemistry - see Ortho-, meta-, para-.---(Para), a state in northern Brazil. 1248 thousand km2. Population 5.1 million people (1991). Adm. c. - Belen.
S.I. Ozhegov, N.Yu. Shvedova Explanatory dictionary of the Russian language
Meaning:
PAIR, -y, w.
1. Two homogeneous objects, used together and forming a whole. P. oars. P. boot.
2. Two pieces of something. (simple). P. apples.
3. Men's suit (trousers and jacket or frock coat, tailcoat). Sew a new pair. Tailored p.
4. Two horse team. Ride in pairs.
5. Two persons located, acting together, united by something. general Go~mi. Dancing item. Married item. Paired with someone. be, work(together).
6. In certain expressions: a person who is part of such an association or suitable for it (colloquial). Not n. who to whom.(does not fit, does not correspond). Find (look for) a mate. To be left without a partner.
7. A small amount of something, several (simple). Call for a few words(to talk a little). For a few minutes.
Meaning:
Y, and.
Two homogeneous or identical objects, used together and forming one whole.
A pair of stockings. Two pairs of boots.
One of these two objects (in relation to the other).
A boot was found under the same bush, which turned out to be a pair of boots found in the bedroom. Chekhov, Swedish match.
2. Razg.
An object consisting of two identical parts joined together.
A pair of trousers. Two pairs of forceps. A pair of scissors.
3. with definition.
Men's suit (trousers and jacket or frock coat, tailcoat).
Kalinovich, having previously dressed himself in new and very good underwear, put on a pair of tails. Pisemsky, A thousand souls.
Finally, Ivan Pavlych came out - in a new pair, in a hard collar, with a mustache curled like a youth. Kaverin, Two captains.
Two horses harnessed to a carriage, cart, etc.; two horse harness.
Already from early morning, threes and couples with multi-colored ribbons on their arches and in their manes were driving around Ukleev. Chekhov, In the ravine.
A couple was harnessed to the groom, and a single person to the bride. Bunin, Village.
Two beings located, acting together, united by something. general; two considered as a whole.
Couple in love. Married couple. Dancing couples.
In the evening on the city boulevard, it is not uncommon to meet a couple sitting hugging and having sweet conversations. A. Ostrovsky, Travel along the Volga from its origins to Nizhny Novgorod.
Groups, couples, and individuals walked from the skating rink to their homes. Ketlinskaya, Days of our Lives.
There were already a lot of people on the tennis courts. Volodya and Nina played in pairs. Dobrovolsky, Three in Gray Overcoats.
One of two beings that make up such a whole.
I found myself at my desk alone, I didn’t have a partner. Nosov, Vitya Maleev at school and at home.
|| in meaning adv. in pairs.
Together, two by two.
Line up the children in pairs. Walk in pairs.
Some ants came out in whole columns, others returned in pairs and separately. Obruchev, Plutonia.
6. in meaning tale, to whom ( usually with a negative). Razg.
The first moments of love can be so intense that people dare to make spontaneous decisions without realizing the consequences of their behavior. A whirlwind of emotions inspires you, your veins bubble, and it seems that a quick engagement or marriage is the best way to keep the holiday inside you. As practice shows, the opposite is true.
A study conducted at Emory University found that the longer modern couples date before marriage, the more likely they are to stay together. In numbers, couples who dated for three years or more were 39% less likely to divorce than those who dated for less than a year.
Lavish preparation for a wedding and a lordly celebration on the occasion of marriage have their own long-standing traditions. Those around the newlyweds, and sometimes the bride and groom themselves, try to throw a party no worse than their neighbor’s, because they are definitely getting married once in their life! But the catch is that partying on a grand scale can come back to haunt you in the future.
In the same Emory University study, researchers surveyed thousands of heterosexual couples and found that “the length of marriage is associated with spending on engagement rings and wedding ceremony.” In particular, those who spent significant sums on rings broke up 30% more often.
And there is a simple explanation for this: dried up gold reserves and debts undermine the financial base of the new unit of society. Money becomes a source of disagreement, economic disputes do not subside, which leads to a split in the family.
Couples who sleep together are happier than those who sleep apart. Scientists from the University of Hertfordshire studied the situation of sleeping spouses and found that 94% of couples who spent the night in contact considered their relationship happy. At the same time, only 68% of those who did not touch each other in their sleep were satisfied with their relationships.
puhhha/Shutterstock.comHugs are also good for the health of your spouse. They will warm bodies in a cold bed in the cold winter.
A simple “thank you” strengthens bonds. Scientists from the University of California at Berkeley say this. The researchers created conditions in which both halves received gratitude from their partner. At the end of the experiment, all 77 couples felt calmer and more satisfied. They understood each other better and felt more keenly the care and responsiveness of their loved one. And there was a scientific basis for this. The positive effect of “thank you” was due to increased production of oxytocin, the hormone of trust and affection.
The health of even the strongest relationships needs to be improved. Experienced people advise declaring their love to each other more often, and scientists advise moving from words to deeds. Moreover, it does not require titanic efforts or large investments from you. It is enough to provide small but extremely significant signs of attention.
Is it difficult for you to prepare a cup of tea for your partner? Nonsense! But it is precisely such little things that strengthen your union. Sociologists from the Open University of Great Britain came to this conclusion. They studied the lives of 5,000 people for two years and found that unexpected surprises and small pleasantries continue to matter even after years of long-term communication. It is a powerful glue in a relationship.
How many times has the world been told that honesty is the best way to solve a problem? But people routinely turn a blind eye to reality, replacing the truth with far-fetched excuses. And they do this, of course, in vain. Psychologists advise talking directly about your preferences and desires.
Researchers at the University of Illinois tracked the relationship developments of 232 couples and noted that successful couples remembered all the details of their romance, while struggling couples lied to themselves, looking back and thinking that everything was fine.
It has long been noticed that people who are members achieve better career success than those who are single. They have better health and social connections, and a more stable psyche. At least run and sign for the sake of a new position and longevity!
But there is no need to rush, because an ordinary civil union is no worse than an official marriage. All the same benefits, but without the documented obligations. Scientists say that the beginning of a life together is not much different between spouses and cohabitants. Moreover, all differences completely dissolve after the honeymoon is over.
In each of us there lives a romantic who wants to find that same soul mate on his life’s path. Some people struggle with it, realizing that there are no complete coincidences, while others idealize their soul mate, believing that it was destined to be so. The latter make a significant mistake.
This is evidenced by sociological studies by Spike W. S. Lee and Norbert Schwarz. According to scientists, people who view their relationships as the dictates of fate, thereby ingraining in themselves the basis for future disappointment, because reality always contradicts fantasy. It is correct to compare yours with a long journey full of obstacles and difficulties. In this case, the passing years will not leave you with the bitterness of unfulfilled hopes.
The further apart the magnets are, the weaker the connection between them. It is precisely this logic that often undermines people’s faith in long-distance relationships, and they break up without ever testing their feelings for strength. “Bye, it was cool, let’s chat on Facebook.”
Wait, wait, don’t rush to run away. Experts at Cornell University found many examples of geographically distant relationships in which men and women felt greater satisfaction compared to living side by side.
mangostock/Shutterstock.com Psychologists say that in long-distance relationships, people more often turn to the bright moments of their lives, savor pleasant details, and this fuels feelings.
Flowers of life. We are told this on national television and when visiting our grandmother. It’s clear that the state needs new citizens, and the older generation wants to shake things up a little. Blindly follow the lead?
Sociological research does not provide a clear answer. Some of them claim that families with children are happier. Others, not without reason, point to the increased level of stress associated with adding to the family, which not everyone can go through. The conclusion is simple: you need to want a child or grow to the proud title of “parent”.
Long-term alliances are built on the principles of mutual respect, love, help and, most importantly, kindness and generosity. This is the opinion of the famous American couple John and Julie Gottman. They support their opinion with forty years of experience as family psychologists. Based on their extensive experience, the Gottman spouses recommend paying attention to your behavior during quarrels. During the period of domestic passions, it is very easy to offend your partner and cause him contempt - the main factor in the impending separation. By understanding that your “fi” can be expressed with good notes, you will not harm your union.
