In nature and technology, oscillations called harmonic are widespread.

Harmonic oscillations are those that occur under the action of a force proportional to the displacement of the oscillating point and directed opposite to this displacement.

You already know that under the action of such a force, a spring pendulum oscillates, therefore, under certain conditions, they can serve as an example of harmonic oscillations (in particular, provided that they are not noticeably affected by the friction force).

With the help of the experiment shown in Figure 63, we will find out by what law the coordinate of an oscillating spring pendulum changes over time and what the graph of this dependence looks like.

Rice. 63. Experience in investigating the time dependence of the coordinates of a spring pendulum that oscillates

In this experiment, some small massive vessel with a small hole at the bottom (for example, a funnel) is taken as a load, and a long paper tape is placed under it. A vessel with sand previously poured into it (or poured with a coloring liquid) is set into oscillatory motion. If the belt is moved at a constant speed in a direction perpendicular to the plane of oscillation, then an undulating path of sand will remain on it, each point of which corresponds to the position of the oscillating load at the moment when it passed over it.

Figure 64 shows the resulting curve. It is called a cosine wave (from a high school mathematics course, you will learn that similar graphs have functions like y \u003d sin x and y \u003d cos x with a variable x). The time axis t is drawn through the points corresponding to the equilibrium position of the pendulum, and the displacement axis x is perpendicular to it.

Rice. 64. Graph of the dependence of the coordinates of an oscillating spring pendulum on time

It can be seen from the graph that the largest deviations of the load from the equilibrium position in both directions are the same in absolute value and equal to the amplitude of oscillations A.

The pendulum began to move from extreme point with coordinate x \u003d A. For a time equal to the period T, the pendulum made a complete oscillation, i.e., having passed the equilibrium position, it reached the opposite extreme point with the coordinate x \u003d -A, lingered for a moment in it, changing the direction of speed to the opposite , then went in the opposite direction and, having passed through the equilibrium position for the second time, returned to the same place from which it began to move. Then the next swing begins, and so on.

If during the experiment the time interval t was measured, during which the pendulum made the oscillations shown on the graph, then their period T can be determined by dividing this time by the number of oscillations: T \u003d t / N. Knowing the period, one can find the oscillation frequency: v = 1/T.

The graph makes it possible to approximately determine the coordinate of the load at any time. For example, through T from the moment of the beginning of the first oscillation, the load was at the point with the coordinate x 1 .

If the graph of the dependence of the coordinate on time of any body is a sinusoid (cosine wave), that is, if the coordinate changes with time according to the sine (cosine) law, then in this case they say that both the coordinate and the body itself perform harmonic oscillations.

• Periodic changes in time of a physical quantity that occur according to the law of sine or cosine are called harmonic oscillations

Figure 65 shows an experiment similar to the one discussed above, only for a filament pendulum. With the help of this experiment, it can be shown that for a filament pendulum, the graph of the dependence of the coordinate on time is also a sinusoid, i.e., that its oscillations are harmonic.

Rice. 65. Harmonic vibrations of a thread pendulum

Theoretically, the oscillations of a filament pendulum would be strictly harmonic if it were a material point oscillating without friction with a small amplitude 1 at a distance from it to the suspension point that does not change with time. (It can be proved that only under these conditions will the force that returns the point to its equilibrium position be directly proportional to the displacement, as a result of which the oscillations will occur according to the harmonic law, that is, according to the law of change of the sine or cosine.)

• A material point that oscillates at a distance from the suspension point that does not change with time is called a mathematical pendulum.

A mathematical pendulum is an abstract model, there are no such pendulums in reality.

In practice, vibrations close to harmonic are made by a heavy ball (for example, steel), suspended on a light and low-stretch thread, the length of which is much greater than the diameter of this ball, with a small amplitude and low friction.

When a body performs harmonic oscillations, not only its coordinate, but also such quantities as force, acceleration, speed, also change according to the sine or cosine law. This follows from the laws and formulas known to you, in which the indicated quantities are directly related in pairs proportional dependence, for example F x \u003d -kx (Hooke's law), and x \u003d F x / m (Newton's second law). From these formulas it follows that the force and acceleration reach highest values, when the oscillating body is in the extreme positions, where the displacement is greatest, and are equal to zero when the body passes through the equilibrium position. This means that the oscillatory motion near the middle position of the body is closest to uniform, and near the extreme positions it differs greatly from uniform motion. The speed, on the contrary, in the extreme positions is equal to zero, and when the body passes through the equilibrium position, it reaches the highest value.

Questions

• According to Figure 63, tell about the purpose, the order of execution and the results of the experiment depicted.
• What do the segments OA and OT correspond to on the graph (see Fig. 64)?
• What vibrations are called harmonic?
• What can be shown with the help of the experiment depicted in Figure 65?
• What is called a mathematical pendulum?
• Under what conditions will a real filament pendulum oscillate close to harmonic?
• How do the force acting on the body, its acceleration and speed change when it performs harmonic oscillations?

1 Recall that a small amplitude means such an amplitude at which the trajectory of the pendulum can be considered rectilinear. Numeric value amplitude that satisfies this condition depends on the accuracy of the result required in the problem being solved. In most practical problems, the amplitude can be considered small if the deflection angle does not exceed 8°.

” and subsection “ ” of the next article ““. In the previous article “How to learn to draw? Easily! »we decided on what drawing is in general - and why it is needed in particular. And also with the fact that the first step in any image of something is setting a goal. In this article, let's touch on the topic of choosing a target for drawing in more detail.

A few tips on choosing the best drawing target will help you refine your drawing target. Please note: about what to draw I won't write anything. This is a personal desire of everyone 🙂 The only thing is that it is better to start with a simpler one - where there are fewer lines, simpler form. Thus, the principle of gradualness in learning is observed. And much easier to learn.

Whereas we will dwell on what to draw and on what to draw in more detail, because this drawing component greatly influences the choice of target.

As mentioned in the first step (Setting a goal), there is a huge amount of what you can draw with and what you can draw on. But they can all be combined, to find a common denominator:

• Draw on the plane.
• Draw with the help of what leaves a mark on the plane.
• The trace can be a point, a line, a plane.

Everything else is additions and development of these points. So, in essence, no matter what you want to start drawing. It is important to determine what exactly in order to prepare for the next step - buying drawing tools. And with a clearly defined goal, it is much easier to work.

The second aspect of refining the purpose for drawing is "why to draw".

This point is necessary in order to distinguish between two such things. There are two huge areas that can be called

1. "academic art" and
2. "drawing for the soul"

They differ very much. First of all, because in academic drawing great attention is paid to the technical side and accuracy.

Whereas when drawing for the soul, something else is most often important.

Let's look at the question from a third perspective. Yes, there are any images and inner experiences. And most often I want to draw some image in such a way that it expresses a personal inner experience. To convey to others what is inside, with the help of a drawing of the surrounding world. Inside is joy - and it is transmitted in the form of a bright landscape (images of nature). Inside grief - and there are dark and gloomy pictures.

And the soul becomes easier.

So, most often there is not up to accuracy - unlike the first area, academic drawing.

If you are only interested in the academic way of depicting, then there is huge mass literature and numerous circles. I will not duplicate this information. Except just a little. For order and comparison.

Further steps of our lessons on provide learning to draw for the soul.

So, if you are serious about learning how to draw, then I advise you not to move on to the next paragraph until you have completed the first step, setting a goal, with the help of the second, clarifying your plan. That is, that why do you want to draw.

Examples of why you might want to draw:

• to make mom happy
• to brag to your friends
• to capture your experiences
• in order to accurately convey what he saw
• to express your thoughts about something
• just for the soul
• to practice creativity
• for company
• and so on and so forth.

As you can see, "why draw" can be a HUGE amount. And the more precisely you decide on why you are drawing, the better you will be able to draw it. Not without practice, of course. But nonetheless.

In our further lessons, we settled on this option:

1. Goal: draw - animals.
2. What and on what to draw - with a pencil or pen on paper.
3. Why draw - for your own pleasure. Well, brag a little.

Having decided on the goal, you can proceed to the next point -

Each person would like to know the secrets of his character and see himself from the outside. in the best way for self-knowledge is a psychological projective drawing that will reflect all the features of your personality.

When we do something that we don’t think about, the result shows not the logical, but the emotional side of the brain. Our experiences, joy, anger and everything else can erupt unconsciously projected on paper.

You will need privacy to take the test below. No one should distract you from the process. Take piece of paper, preferably A4, handle or pencil. Divide the sheet into equal four parts and number them so that it looks something like this:

The final preparation of the sheet will look like this:

• in the first sector, draw a dot in the middle
• in the second sector draw a small square or rectangle
• in the third sector there should be something similar to the "=" sign
• the fourth sector remains empty

You should get something similar:

So, free your thoughts, there will be no difficult tasks - draw only what comes to your mind right away.

In the first three sectors, just draw what you want. Draw what you feel without thinking. You don't need to draw something complicated - spend about 10 seconds per sector. After that, move on to the last task.

In the fourth sector, draw a boat that floats on the waves. The drawing should also be primitive - also do not spend a lot of time on it.

Deciphering the drawings

Sector one

This section represents your isolation or openness. If there are round figures in this sector, then you are a closed person. The same applies, for example, to a drawn spiral leading to the center (point). If you just drew a lot of dots, then you are very sociable and open. Roughly speaking, the more elements you applied, the better. The only exception is circles. They should be as few as possible.

Sector two

This section shows your kindness and responsiveness, as well as the level selfishness. You may consider yourself kind person, if what you drew is connected in some way to the original rectangle. The smaller the contact area, the more selfish you are. Open people often draw a wall, which indicates their focus on the company and the team. Many draw a house, which is also very good. If your drawing does not touch the rectangle, then this indicates that you need to think - maybe you are selfish.

Sector three

This sector will show how you consider yourself successful or successful with the opposite sex - is it easy for you to communicate and flirt. There should be as few drawing elements as possible and they should not touch the equal sign (as most people call it). People who are satisfied with their relationship with the opposite sex usually draw a smiley face to make something like "=)". Worst of all, if the drawing turns out to be symmetrical and heavily piled up, this is an indicator of your self-doubt in communicating with the opposite sex.

Sector four

Everything is extremely simple here. This sector shows you what you are in love and in relations. If the sea turned out to be restless, then you prefer dynamic love that breaks your head. If the sea is calm, then love for you is calm, warm and only pleasant, ordered emotions. As for the boat itself, you should pay attention to whether it is difficult to draw, as well as what is happening around it. Any excesses like sails or people (which were not discussed), stars, living creatures, the Moon, the Sun - speak of your romance. If none of this is there, then you are strong in character, and romance is alien to you.

Thus, the simplest at first glance test can show each person his true face. Psychological personality tests of this kind are the most accurate, since the test subject does not need to think - he is driven only by the emotional part of the brain. We wish you good luck and don't forget to press the buttons and

Incredible Facts

We often unknowingly scribble when we're on the phone or sitting in a boring meeting. We may not understand it, but these doodles actually have a lot to say about personalities and mood of the painter.

The size, position, and object of the drawing give us some key clues. In the case of doodles, as in the case of dreams, there are some that occur quite frequently and as such are easier to interpret.

Size and Placement

Large scribbles that take up most of the page show that you are drawn to nature. In addition, this may be a signal that you are closed in on yourself, but you want to communicate.

Small drawings in the corner of the sheet indicate that the person is neat and organized in his own way. Finding scribbles on the sides of a sheet is a sign that a person likes when everything is in its place.

Those who place their drawings in the center of the page are the extroverts most often in need of attention. it common feature people working in areas where you need to constantly speak in public (lawyers, teachers, various officials etc.). The center of the page is also a need for personal space.

The scribbles at the top of the page demonstrate confidence and an abundance of ideas. If the drawing is drawn right next to the heading of the text on the sheet, this means that the "artist" does not consider what is written on paper important.

The left side of the sheet is the most common place where scribbles are usually located. This speaks of nostalgic feelings for the past in a person. Usage right side the sheet is not so common, mostly left-handers draw on the right. This may indicate a desire to express oneself or reveal secret thoughts.

What do unconscious drawings mean?

Objects and items

So what exactly are you drawing?

Animals

Those who draw animals are usually very sensitive to any living beings. The type of animal that a person draws speaks of his mood, and, often, the type of personality that he would like to have.

Read also:Love to draw doodles? Google will hire you

Drawings of small, weak and passive animals indicate slowness, lack of self-confidence, and also that a person is more of an introvert. Images of aggressive animals speak of self-confidence, funny and cute animals are the playful nature of the painter, and slow animals are a brazen, contemplative personality.

Separately, it is worth mentioning the elephant. He is powerful and dominant, demonstrating the inner strength of a person that can overcome any obstacle. Elephant painters are often funny and a little mischievous.

Arrows

Arrows are a drive and a high degree of motivation. These people are aggressively ambitious.

chains

Chains tend to indicate that a person has feelings of limitation, especially in a relationship or at work. Restrictions can also refer to a person's beliefs that prevent him from doing what he wants. If a person appears in one of the links of the chain in the figure, then this indicates a very tense state.

Circles

Circles are a need to find unity and peace. They can indicate a person who is trying to collect all the pieces of the puzzle and understand everything.

They can also speak of a strong intuitive feeling that the pieces of the puzzle must fit together to form a whole. It doesn't matter if it's about relationships or life in general.

Circles also symbolize lighter feelings compared to corners. They are associated with sociability, talkativeness, friendliness, as well as the desire to be flexible and loving.

The meaning of unconscious drawings

Clouds

Fluffy clouds indicate that the painter is happy, he is a little dreamy, but possesses strong feeling freedom. "Angry", clouds before rain, are depression or difficulties in everyday life.

Cubes

Cubes indicate a person's desire to be constructive. If they are three-dimensional, then this indicates the ability or attempt of the painter to see the problem from all sides. Drawing cubes is often a sign that the person feels trapped and powerless in the face of the situation.

In combination with circles and spirals, the cubes indicate that the painter would actually prefer to be in a different place now, doing something more interesting.

Boxes and cubes are the most common drawings that people unconsciously draw.

It is important to add that boxes, squares and other three-dimensional objects are an indicator of masculinity. Moreover, if you draw boxes, you most likely have a practical, organized and methodical approach to various things.

points

The dots speak of anxiety and instability.

Ears

They testify to the need to listen to someone for too long and too much.

Eggs

Eggs symbolize a new beginning. There may be a new talent that you didn't know about. Eggs can also indicate that you have a need and desire to settle down and build your own stable, secure nest.

Drawings that we draw unconsciously

Eyes

Eyes often speak of the inner self of the painter. Big eyes draw open people, and small - closed. Closed eyes indicate a refusal to look within. Sad or happy eyes are a reflection of how a person feels deep inside.

Eyes are often drawn by people who want to be attractive. Many painters immediately assume that perhaps the drawing of the eye is associated with their feeling that they are being watched. In severe cases, the eyes may indicate a person's paranoid tendencies.

faces

The expression of a painted face is a good indicator of the mood and character of the person who painted it. well drawn Beautiful face says that you see the good in other people.

If you draw strange or ugly faces, then this indicates your incredulity. Comic faces are the desire to be the center of attention. Baby faces are the desire to feel needed. Drawing a human profile indicates that you are an introvert.

Flowers

Flowers are the feminine side of a person and the desire to see growth, nature and procreation. If the painted flowers are a harmonious bouquet, then a person is attached to his family and feels complete unity with her. Jung believed that if flowers appear in a dream, then this is due to the need to release emotions that a person cannot express openly.

Unconscious drawing of flowers is femininity. This is the reason why most often it is girls who draw them. In addition, flowers are an understanding of one's personality and the expectation of flowering.

Food

The image of food or drink suggests that a person is hungry for feelings and needs love.

hearts

Hearts say that in this moment a person experiences romantic feelings.

Houses or buildings

They reflect the attitude of a person to his family life. Houses must have doors and windows that indicate that a person can see what is happening outside, and his loved ones can see and hear him.

A warm, hospitable house that seems lived-in, perhaps with smoke curling from the chimney, indicates that the painter is completely satisfied with him family life and he is happy.

Chaotic, gloomy houses speak of difficult associations for a painter with home life. The house depicted on top of the hill indicates that you are very lonely.

Also, drawing houses shows that you attach a lot of importance to shelter and security. What you paint on houses speaks volumes about the things you value the most. For example, numbers and dollar signs indicate your preoccupation with money.

Airplanes, cars, ships and other vehicles can indicate your desire to travel, change something in your relationship or change your life.

If you are drawing a simple house with no doors, no windows, no curtains, no chimney with smoke, you are most likely very upset at this point in your life, feel lonely and would like to talk to someone.

On the other hand, if you draw a big house, with flowers and a garden around, you are almost certainly happy. It can also indicate that the person is materialistic and loves luxury.

Some draw thin and tall houses. This indicates a critical mind. Such people do not show an open approach to various things.

Names and initials

Drawing your name or initials is common for people who like to be the center of attention. Teenagers often draw only their first name, without a last name, thereby demonstrating their desire to break away from the family and do their own thing.

On the other hand, drawing someone else's name says two things: either it's the name of a person you have romantic feelings for, or it's the name of a person associated with a problem that you need to deal with.

lines

Some people just draw lines. If at the same time they strongly press the handle, then they are overcome by aggression, and they are afraid of something. This pressure determines your mood. The lighter you move your pen across the sheet, drawing the lines, the more calm you have inside.

People

Drawing only the profile of a person indicates that the painter does not like the way he or she looks, or the person realizes that he or she cannot draw well. In full drawings, but when, for example, a drawn person has no hands, this indicates a feeling of being unable to do anything.

The lack of legs is the feeling of having no roots, even if there are roots. It is important how a person painted a face, it speaks of his emotions. The missing parts of the face speak of a person who does not feel complete, whole.

Emphasizing and drawing only the eyes speaks of the caution and suspicion of the painter. Drawing ears, as a rule, speaks of the need to listen to someone for too long.

Child's drawing: if a drawn person is floating in the air, then he means a lot to the baby. Very large ears that are not adorned with earrings may indicate verbal abuse. Sometimes the lines leading to the ears exhibit a commonly encountered offensive tone.

A mouth covered with a long slit can be drawn by an anxious child, and a complete absence of a mouth is an inability to communicate. Empty eyes speak of difficulty in perceiving the surrounding reality, they are blind to what is happening.

Big teeth irregular shape- This is a tendency towards aggressive behavior, especially in combination with other jagged graphic symbols, such as "spiky" fingers, legs, ears, hair.

Interesting fact: children draw feelings that they cannot express.

Stars

The stars speak of a sense of hope, anticipation of something, optimism. People who paint stars in groups are irrepressibly romantic.

Web

It usually symbolizes the feeling that either the person has fallen into a trap, or wants to lure someone into a certain relationship or situation.

Transport

Drawings of any type of transport speak of a desire to get away from something or to achieve a goal. The faster the type of vehicle, the higher the speed at which the person operates.

triangles

Triangles are the second most common unconscious drawing. They speak of a rational state of mind and a desire to know what is going to happen.

Trees

Trees represent our egos and our ambitions, so pay attention Special attention to the health of the tree! Trees with leaves and fruits show that the painter considers love, sexual relations and children to be part of a single whole.

Bare, hanging branches indicate depression and lack of morale. Barren trees may indicate that the painter does not feel his roots.

The tree also indicates a person or thing from the past that you cannot forget. Moreover, you are afraid of getting into trouble and want to be protected if your tree stands alone. If the tree is surrounded by flowers, it indicates happiness and love for the family.

Zigzags

Zigzags are a discomfort in life that you want to run away from. They can also talk about the energy of a person who wants to learn how to get along with others. If the zigzag has soft, flowing lines, it speaks of a romantic, feminine approach to things. Zigzags drawn with a large number of straight lines indicate a more aggressive masculine nature.

What do the drawings that we draw unconsciously mean?

Colors

When people unconsciously draw, they usually do it with what they have at hand. Depending on cultural differences, the same colors can mean different things.

On all human activity lies the imprint of his personality, and to the greatest extent its features are manifested precisely in creativity. Looking at paintings at an exhibition or in art gallery, we see not only a work of art, but also the inner world of its creator, we are trying to understand what exactly prompted the author to write this or that canvas.

Drawing is nothing more than a reflection of thoughts, feelings, attitude and character of a person, it serves as a means of penetration into the spiritual space of a person.

What will the drawing tell about?

Psychological analysis of a person's personality according to his drawings as a method of professional psychology arose at the beginning of the 20th century. Then the first tests appeared, the purpose of which was to diagnose various aspects of the personality according to the drawings of a person.

Nowadays, psychologists have no doubts about what great importance has a drawing in psychology, because it is the drawing that is the most important source of knowledge about the human psyche.

The psychological “decoding” of the drawing gives a qualitative idea of ​​the personality of an adult, it is almost impossible to “fake” or predict the result, because the diagnosis and interpretation of the drawing is a complex and multi-layered process, somewhat similar to the definition.

Doctor of Psychology, Professor Elena Potemkina conducted a large-scale psychological study in the period from 1985 to 1995. This is the analysis and interpretation of more than 100,000 drawings that have been made by people. different ages and spheres of activity, which were at the same time in different life circumstances and mental states.

Thus, the professor managed to draw up a general model for the analysis of the drawing, with the help of which it becomes possible to qualitatively diagnose and interpret the features of human psychology, to determine his character from the drawing.

The table presents general features by which individual personality traits can be determined.

General Pattern Analysis Model

Psychological diagnosis using a drawing in psychology is called projective diagnosis. This name is explained by the fact that in psychology a drawing is an individual projection inner world person, his character and personality traits. Psychology calls tests according to human drawings projective tests or methods, they are of key importance among other methods.

There are many interesting projective techniques, such as "Family Drawing" or, which can tell a lot about the nature of the creator of the drawing.

What exactly do the drawings of a person say?

You will learn about this through the example of interpretation projective technique"Self-portrait".

Of course, high-quality diagnostics should be performed by a professional psychologist using a wide range of different methods, but you and I know how interesting it is to “diagnose” yourself and your loved ones on your own, to learn something new and interesting about yourself and others. You can find many different tests on the Internet, but are you sure of their authenticity and effectiveness?

That is why we bring to your attention Full description professional projective technique "Self-portrait" adapted by the American researcher R. Burns with additions by E. Potemkina.

Paint your self-portrait

The test task is quite simple and sounds like this: “Take a simple pencil or pen, a sheet of paper, 10x15 in size, and draw on paper yourself, alone, and possibly with your family or colleagues. Your artistic ability doesn't matter."

Interpretation by drawing elements:

Head

A big head speaks of a person’s great intellectual claims, or vice versa, of dissatisfaction with the development of the intellect. A small head speaks of the inadequacy of judgments regarding one's intellectual or social capabilities.

Eyes

Large eyes characterize suspicion, as well as a strong dependence on public opinion. small or closed eyes talk about increased focus on one's personality or the closeness of a person.

Ears and nose

Enlarged ears indicate a high sensitivity to criticism. If the nose is highlighted as an outstanding accent of appearance, this indicates sexual problems. Strongly defined nostrils are evidence of aggression.

Mouth

A prominent mouth may indicate speech problems. If the mouth is not drawn at all, then this means a negative mental state, possibly depression, or simply lethargy in communication.

Arms

They are a symbol of human contact with society and the world as a whole. Shackled hands speak of rigidity, isolation, increased demands on oneself. Sluggishly lowered hands speak of the general inefficiency of a person. Weak, fragile hands suggest self-doubt. Long and strong hands on the contrary, they speak of strength and ambition. Very short arms are evidence of a feeling of inadequacy.

Legs

Large legs mean the need for security. The longer the legs, the more expressed the desire to be independent. The absence of legs suggests instability, a feeling of emptiness, lack of a life foundation.

Interpretation by image type:

• Schematic image. The image in the form of general details and schemes belongs to people of an intellectual warehouse, "thinkers", for whom the most important role play general information about different phenomena.
• Realistic image. This is a detailed drawing of the appearance, characteristic of a pedantic person, prone to deep analysis and constant refinement.
• Image metaphor. A person draws himself not as a person, but as some object, creature or phenomenon, for example, an animal, rain, mug, movie or book character. This speaks of developed creativity, artistic warehouse, developed imagination and creativity. Such a person has a good sense of humor.
• Portrait in the interior. A person draws himself against the backdrop of a landscape, in certain circumstances. surrounded different items. This speaks of extraversion, the desire to be surrounded by other people.
• Emotional portrait. Picture yourself in some emotional state. This indicates a high emotionality and sensitivity, a tendency to deep self-reflection. Often a person draws the opposite emotion: he is sad, but depicts himself with a smile on his face.
• Image of a portrait from the back. Instead of a face, a person draws a back of the head, which indicates an internal contradiction, a desire to demonstrate one's own. This may also indicate a reluctance to depict certain details, such as a face.
• Aesthetic image. characteristic of people with developed artistic ability and aesthetic taste.

A small plot "What do your drawings say?"