As you already know, a general way to obtain carboxylic acids can be the oxidation of the corresponding aldehyde according to the equation
In industry, this process is carried out using atmospheric oxygen in the presence of catalysts.
Analyzing general formula carboxylic acids, you can see that its molecule contains two parts - a hydrocarbon radical (alkyl) and a functional group
which is called carboxyl. The name of the functional group comes from the names of the carbonyl >C=0 and hydroxyl -OH that form it.
The class of carboxylic acids is extremely diverse. It includes one-, two- and polybasic acids, unsaturated and aromatic. But these subclasses of carboxylic acids are the subject of study in schools and science classes. We will just mention two unsaturated acids: C 17 H 33 COOH - oleic acid (contains one double bond in the molecule) (Fig. 50) and C 17 H 31 COOH - linoleic acid (contains two double bonds in the molecule). These acids are called fatty acids, and now, obviously, you understand why we draw your attention to these acids - they are part of liquid fats.
Rice. 50.
Scale model of the oleic acid molecule
However, let us return to the limiting monobasic carboxylic acids. The homologous series of these acids begins with a compound that does not fully meet the above definition - formic, or methanoic acid.
As you can see, the carboxyl in its molecule is connected not to a hydrocarbon radical, but to a hydrogen atom, like the carbonyl in formic aldehyde (Fig. 51).
Rice. 51.
Model of the formic (methanoic) acid molecule:
Obviously, the names of acids and their corresponding aldehydes are identical.
The fact that the structure of formic acid differs from the structure of the molecules of other monobasic carboxylic acids also determines the peculiarities of its chemical properties. It undergoes a “silver mirror” reaction like aldehydes, since its molecule is a synthesis of two functional groups: carbonyl and carboxyl.
Formic acid is a liquid with a pungent odor (t boiling point = 100.8 ° C), highly soluble in water. Formic acid is poisonous! If it comes into contact with skin, it causes burns. The stinging liquid secreted by ants, nettles, and some types of jellyfish contains this acid (Fig. 52).
Rice. 52.
The stinging liquid contains formic acid:
1 - jellyfish; 2 - nettles; 3 - ants
Formic acid has a disinfectant effect and is therefore used in food, leather and pharmaceutical industry, as well as in medicine. In addition, it is used for dyeing fabrics and paper (Fig. 53).
Rice. 53.
Application of formic acid:
1 - leather industry; 2 - fabric dyeing; 3 - medicine
Acetic or ethanoic acid
(Fig. 54) is a colorless liquid with a characteristic pungent odor, miscible with water in any ratio. Aqueous solutions acetic acid go on sale under the name “table vinegar” (3-5% solution), “vinegar essence” (70-80% solution) and are widely used in Food Industry.
Rice. 54.
Model of the acetic (ethanoic) acid molecule:
1 - ball-and-rod; 2 - scale
Acetic acid - good solvent many organic compounds, used in dyeing, leather production, and the paint and varnish industry (Fig. 55). In addition, acetic acid is feedstock for the production of many technically important organic compounds: artificial fibers, pesticides, film and photographic films, etc. Acetic acid is extremely dangerous if it comes into contact with the skin, so safety precautions must be observed when working with vinegar essence.
Rice. 55.
Application of acetic acid:
1 - canning; 2 - production of artificial fibers and fabrics; 3 - seasoning for food; 4-8 - production of organic compounds (4 pesticides, 5 varnishes, 6 paints, 7 photographic films, 8 glues)
With increasing relative molecular weight in the homologous series of saturated monobasic carboxylic acids, their density, boiling and melting points increase, and solubility in water decreases.
Higher carboxylic acids, also called bold (guess why), are solids. These are, for example, palmitic C 15 H 31 COOH acids (Fig. 56, 1) and stearic C 17 H 35 COOH acids (Fig. 56, 2).
Rice. 56.
Scale models of molecules:
1 - palmitic acid; 2 - stearic acid
The chemical properties of carboxylic acids are determined primarily by their belonging to the type of acid in general. Like inorganic acids, carboxylic acids are electrolytes, albeit very weak ones, and therefore dissociate reversibly:
Aqueous solutions of carboxylic acids change the color of indicators.
As the hydrocarbon radical increases, the degree of electrolytic dissociation decreases.
Like inorganic acids, carboxylic acids interact with metals, basic and amphoteric oxides, bases, amphoteric hydroxides and salts.
Thus, formic and acetic acids interact with metals in the electrochemical voltage series up to hydrogen:
These acids react with basic and amphoteric oxides to form salts - formates and acetates:
Similarly, formic and acetic acids interact with bases and amphoteric hydroxides:
These acids interact with salts of weaker acids. Reactions proceed to completion if a precipitate or gas is formed:
Organic acids, as you already know, undergo esterification reactions with alcohols, forming esters, according to the equation
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
Goals:
Equipment and reagents: acetic acid, formic acid, universal litmus paper, methyl orange, viburnum aqueous extract, copper wire, iron (III) hydroxide, sodium bicarbonate, silver nitrate, ammonia(for preparing an ammonia solution of silver oxide), potassium permanganate solution; test tube holder, matches, alcohol lamp, test tubes; demonstration posters, demonstration preparations (Kinder Surprise eggs), multimedia, video experience (solubility of carboxylic acids in water, interaction of acetic acid with metals), natural objects (lemonade, ketchup), front work sheet (FPR) ( Annex 1 ).
Methods: verbal-visual, laboratory experiment, group work.
Methodology: traditional lesson, learning something new.
DURING THE CLASSES
- Good afternoon!
“Every substance - from the simplest to the most complex - has three different but interconnected aspects - property, composition, structure.”
V.M. Kedrov
One of the leading ideas in the science of chemistry is the dependence of the properties of substances on their composition and structure, which we have to study and confirm today.
Today's lesson is devoted to a special class of organic compounds. Which one? Let's plunge into the past.
Since ancient times, people have grown grapes and stored grape juice for future use. During storage, the juice fermented, resulting in wine. If the wine turned sour, vinegar was formed. This explains the origin of the word “vinegar” - from the Greek “oxos” - sour. (Demonstration of the “table vinegar” drawing).
People began using vinegar almost 3,000 years ago. In ancient times, vinegar was the only food acid. Later, an important “additive” to various culinary products appeared - citric acid... For the first time it was isolated from the juice of unripe lemons.
To give food a sour taste, sorrel leaves, rhubarb stems, lemon juice, and sorrel berries were used. (demonstration of drawings). Of course, then no one thought that the sour taste in all cases is due to the presence of compounds of the same class. Which one? (Acids) .
Organic acids, which are called carboxylic acids.
Cranberries, lingonberries, blueberries and honey contain benzoic acid. It is widely used in the food industry as a preservative (E210) in the production of drinks and ketchups. (demonstration of lemonade and ketchup).
Many insects that form families or simply “communities” (termites, ants, wasps, bees) produce special chemicals in their bodies, with the help of which they notify their fellow tribesmen of danger. For example, red ants have an alarm pheromone - formic acid, which also serves as their weapon.
Formic acid is also found in some plants, particularly stinging nettle.
- Guys, formulate the topic of the lesson. (Carboxylic acids). Write down the date and topic of the lesson.
By studying infusions obtained from the roots and leaves of various plants, by the end of the 18th century, Karl Scheele isolated tartaric, citric, malic, gallic, and oxalic acids.
– Today’s lesson is devoted to one of the class of organic compounds – carboxylic acids. You have studied inorganic acids.
– What goal will we set for today’s lesson? (Determine whether carboxylic acids have inorganic properties by studying their composition and structure).
Showing the formulas of some carboxylic acids(Appendix 2 ).
– What do you see in common in their structure? (One or more groups).
– Mentally “split” the unknown functional group into two.
– Which of the studied groups can be found in its composition? (Hydro xylene And carbo nil). Hence the name - carboxyl group.
– Now try to formulate a definition of carboxylic acids.
Carboxylic acids– organic substances whose molecules contain one or more carboxyl groups connected to a hydrocarbon radical.
It is quite obvious that it is impossible to become familiar with all acids. Therefore, let us turn to their classification.
– Looking at the formulas of acids, classify them into groups according to different criteria.
– By what criteria can you divide them? (By the nature of the hydrocarbon radical, by basicity).
(Student's work at the blackboard).
Carboxylic acids (by the nature of a hydrocarbon radical) |
||
| Limit (saturated) | Unsaturated (unsaturated) | Aromatic |
 |
||
Remember the classification of inorganic acids.
What is meant by basicity of inorganic acids? (The number of hydrogen atoms that can be replaced by a metal).
The same is true for organic acids.
Carboxylic acids (by basicity) |
||
| Monobase | Dibasic | Polybasic |
  |
 |
 |
So, carboxylic acids are saturated, unsaturated and aromatic, as well as monobasic, dibasic and polybasic.
In today's lesson we will study monobasic saturated carboxylic acids.
The diversity of carboxylic acids, like all organic compounds, is provided by two important phenomena.
What are these phenomena? (Isomerism and homology).
Formulate a definition of homologues. (Homologues are substances that have the same qualitative composition, but a different quantitative composition (they differ by one or several groups – CH 2), have a similar structure, and therefore similar properties.
On your desks there are pieces of paper that you will paste into your notebooks ( Appendix 3
). The first column contains the first members of the homologous series of carboxylic acids and some of the most common others. In the second you enter their systematic name (Independently with subsequent verification). The third, fourth and fifth columns are already filled.
Now let’s name carboxylic acids according to systematic nomenclature. Let us recall the rules for the nomenclature of organic compounds using an example:
In the systematic nomenclature of carboxylic acids, the ending is used -oic acid. If there are several carboxyl groups, then prefixes are used di-, three-, tetra- etc. Most often, historically arose names are used for carboxylic acids, associated in most cases with the names of their natural sources. Therefore, in order to better navigate the nomenclature of organic compounds in the future, you should remember the names of the simplest monobasic acids.
The first representative is formic acid. It contains only one carbon atom. What hydrocarbon corresponds to it? (Methane) .
So, what will the carboxylic acid be called? (Methanoic acid) .
Let's write down the name.
| Formula acids | Name |
Name acid residue | Formula acid residue | |
Systematic |
Trivial | |||
| HCOOH | Methane oic acid | Ant | formate | HCOO |
| CH3COOH | Ethane oic acid | Vinegar | acetate | CH 3 COO – |
| C2H5COOH | Propane oic acid | Propionic | propionate | C2H5COO – |
| C3H7COOH | Butane oic acid | Oily | butyrate | C 3 H 7 COO – |
| C4H9COOH | Pentane oic acid | Valerian | valerate | C4H9COO – |
| C5H11COOH | Hexane oic acid | Nylon | caprat | C5H11COO – |
| CH 2 =CH–COOH | Propen oic acid | Acrylic | acrylate | CH 2 =CH–COO – |
| C15H31COOH | Hexadecane oic acid | Palmitic | palmitate | C 15 H 31 COO – |
| C17H35COOH | Octadecan oic acid | Stearic | stearate | C 17 H 35 COO – |
We have named only one phenomenon that provides the diversity of aldehydes. Now look . (I show the children 3 Kinder Surprise chocolate eggs). Outwardly they are absolutely identical. We also have three absolutely identical molecular formulas of substances. (3 posters are hung on the board in advance with the closed side). I open them(Appendix 4
).
But it is known that these are 3 different substances. What is the reason? What phenomenon is this connected with? (With the phenomenon of isomerism).
Just as chocolate eggs are identical on the outside but contain completely different toys inside, so these substances have the same molecular formulas, but different structures.
Formulate a definition of isomers. (Isomers are substances that have the same qualitative and quantitative composition, but different structures).
Compose isomers for a substance with the molecular formula C 4 H 8 O 2. (Work at the board).
Other isomers can be formed. Let's stop there.
Conclude what types of isomerism are characteristic of carboxylic acids. (Carbon skeleton isomerism (1 and 2) and interclass isomerism (e.g. 1 and 4)).
What class of organic compounds are carboxylic acids isomeric with? (With esters).
Now let’s name compounds 1 and 2 according to systematic nomenclature. (Work at the board).
Knowing the systematic name, you can create a structural formula. Write the formula for 2-hydroxypropanoic acid. (Work at the board).
This acid has a trivial name - dairy– takes an active part in life processes .
Back in the last century, I.I. Mechnikov noticed that eating lactic acid products acidifies the intestines from putrefactive microflora and promotes longevity.
Let me remind you that the properties of a substance depend on its structure. Let's turn to the structure.
In a carboxylic acid molecule R-electrons of the atom ABOUT–hydroxyl group interact with electrons π
-bonds of the carbonyl group, resulting in polarity O-N connections, is strengthened π
-bond in the carbonyl group decreases δ+
charge on an atom WITH and partial increases δ+
on an atom N. This promotes the formation of strong hydrogen bonds between carboxylic acid molecules.
In the homologous series of carboxylic acids, their strength decreases with increasing hydrocarbon radical, so the strongest of them is formic acid. This is explained by an increase in the positive inductive effect of the alkyl substituent in the series –H > –CH 3 > –C 2 H 5 . Further elongation of the carbon chain does not have a noticeable effect on the value +I-effect and therefore acid strength:
We will describe the physical properties of carboxylic acids by performing LPR No. 4. Let's find out how the physical properties of carboxylic acids change in the homologous series. Let's watch a video experiment - the solubility of carboxylic acids in water.
Let's remember the properties inorganic acids. (Students' answers).
The acidic properties of carboxylic acids are similar to those of weak inorganic acids. Inorganic acids dissociate (soluble in water), change the color of the indicator (soluble in water), interact with metals in the electrochemical series of metal voltages up to H 2, with amphoteric and basic oxides, with bases, and with salts of weaker acids.
Do carboxylic acids have these properties? Let's check.
Let us repeat the safety rules when working with acids.
If solutions of acids or alkalis come into contact with the skin, shake off visible drops and then wash off with a wide stream of cool water; Do not treat the affected area with a moistened swab. Why?
Choose the correct answer.
In case of an acid burn, the skin must be treated with the following solution:
Neutralization of hydrogen cations cannot be carried out with caustic alkali, as it can cause chemical and thermal burns. For this purpose, use a soda solution that has an alkaline environment. Answer: 2.
Let's conduct experiments. (Work in groups with writing the corresponding reaction equations on the board).
Let's look at the rebus. . What acid will we work with?
One of the groups will research special properties formic acid, having completed No. 7 LFR.
Move practical work and expected student responses - Appendix 5 .
Interaction of acetic acid with metals - video experiment.
Will alcohols interact with carboxylic acids? (Yes).
Such reactions are called reactions esterification.
Esterification reaction is the formation of esters by the interaction of acids and alcohols in the presence of a water-removing agent.
In accordance with the mechanism of the esterification reaction, during the formation of an ester, a hydroxyl group is split off from an acid molecule, and a hydroxyl hydrogen atom is split off from an alcohol molecule:
Application of carboxylic acids (slides).
- Let's summarize:
– What new did we learn in the lesson? What have you learned? What do we know?
(Short answers from students).
– Formulate a conclusion about the properties of carboxylic acids. (Carboxylic acids have the properties of inorganic acids and exhibit specific properties).
Soluble acids dissociate and change the color of the indicator, carboxylic acids interact with active metals with the release of hydrogen, they react with basic and amphoteric oxides, bases, and salts of weaker acids.
– Today we have gone from the composition and structure of substances to predicting their properties.
Homework: §30, LFR No. 13, 14, 15.
Bibliography
Section of a cone by a plane
Taking into account the dependence of the position of the cutting plane in the section of the cone of revolution, various lines can be obtained, called lines of conic sections.
If the cutting plane passes through the vertex of the cone, its section produces a pair of straight lines - generators (triangle). As a result of the intersection of a cone with a plane perpendicular to the axis of the cone, a circle is obtained. If the cutting plane is inclined to the axis of rotation of the cone and does not pass through its vertex, an ellipse may appear in the section of the cone (the cutting plane intersects all generatrices of the cone); parabola (the cutting plane is parallel to one of the generatrices of the cone) or hyperbola (in in this case the cutting plane is parallel to the two generatrices of the cone) based on the angle of inclination of the cutting plane (Fig. 39).
Rice. 39
It is known that a point belongs to a surface if it belongs to any line of this surface. For a cone, graphically the simplest lines are generators and circles. Consequently, if according to the conditions of the problem it is required to find horizontal projections of points belonging to the surface of the cone, then one of these lines must be drawn through the points.
Figure 40 gives an example of constructing projections of the section line of a cone with a frontally projecting plane, when an ellipse is obtained in the section.
To construct a curved line obtained when a conical surface intersects a plane, in the general case, the points of intersection of the generatrices of the conical surface with the cutting plane are found. To do this, you can divide the base of the cone into equal number parts (usually 12), draw horizontal projections of the generatrices s1, s2,.... s12 and construct their frontal projections. On the frontal projection, the frontal projections of the points of intersection of the constructed generators with the frontal trace of the secant plane Q are marked. Horizontal projections are constructed in projection connection on the corresponding projections of the generators. The profile projection of the cone section line, plane Q, is constructed using the frontal and horizontal projections of points in the projection connection.
When a right circular cone intersects with a plane, the following second-order curves can be formed: circle, ellipse, hyperbola and parabola. The appearance of these curves depends on the angle of inclination of the cutting plane to the axis of the conical surface.
Below we will consider a problem in which it is required to construct projections and the natural size of the section of a cone ω by plane α. The initial data is presented in the figure below.
The construction of the intersection line should begin with finding its characteristic points. They determine the boundaries of the section and its visibility in relation to the observer.
Through the axis of the conical surface we draw an auxiliary plane γ, parallel to P 2. It intersects the cone ω along two generators, and the plane α along the frontal f γ . Points 1 and 2 of intersection of f γ with the generators are boundary points. They divide the section into visible and invisible parts.
Let's determine the highest and lowest points of the intersection line. To do this, we introduce an additional cutting plane β through the cone axis perpendicular to h 0 α. It intersects the conical surface along the generators SL and SK, and the plane α along the straight line MN. The required points 3 = SL ∩ MN and 4 = SK ∩ MN define the major axis of the ellipse. Its center is at point O, which divides segment 3–4 in half.
To construct the section projections most accurately, we will find a number of additional points. In the case of an ellipse, it is advisable to determine the value of its small diameter. To do this, draw an auxiliary horizontal plane δ through the center O. It intersects the conical surface along a circle with diameter AB, and the plane α intersects horizontally h δ. We construct horizontal projections of the circle and straight line h δ. Their intersection defines the 5" and 6" points of the small diameter of the ellipse.
To construct intermediate points 7 and 8, we introduce an auxiliary horizontal plane ε. Projections 7" and 8" are defined similarly to 5" and 6", as shown in the figure.
By connecting the found points with a smooth curve, we obtained the contour of an elliptical section. In the figure it is indicated in red. The frontal projection of the contour changes its visibility at points 1 and 2, as noted above.
To find the natural size of the section, we rotate the plane α until it aligns with the horizontal plane. We will use the trace h 0 α as the axis of rotation. Its position in the transformation process will remain unchanged.
The construction begins with determining the direction of the frontal wake f 1 α. On the straight line f 0 α we take an arbitrary point E and determine its projection E. From E we drop a perpendicular to h 0 α. The intersection of this perpendicular with a circle of radius X α E"" determines the position of point E" 1. Through X α and E" 1 we draw f 1 α.
We construct a projection of the horizontal line h" 1 δ ∥ h 0 α, as shown in the figure. Points O" 1 and 5" 1, 6" 1 lie at the intersection of h" 1 δ with lines drawn perpendicular to h 0 α from O" and 5 ", 6". Similarly, on the horizontal h" 1 ε we find 7" 1 and 8" 1.
We construct projections of frontals f" 1 γ ∥ f 1 α, f" 3 ∥ f 1 α and f" 4 ∥ f 1 α. Points 1" 1, 2" 1, 3" 1 and 4" 1 lie at the intersection of these frontals with perpendiculars restored to h 0α from 1", 2", 3" and 4" respectively.
Lecture 16. CONE PROJECTIONS
A cone is a body of rotation.
A straight circular cone belongs to one of the types of bodies of revolution.
A conical surface is formed by a straight line passing through some fixed point and successively through all points of some
swarm curve guide line. The fixed point S is called the vertex. The base of the cone is the surface formed by a closed guide.
A cone whose base is a circle and whose vertex S is on the axis
perpendicular to the base passing through its middle is called a right circle
govy cone. Rice. 1.
The construction of orthogonal projections of the cone is shown in Fig. 2.
The horizontal projection of the cone is a circle equal to the base of the cone, and the vertex of the cone S coincides with its center. On the frontal and profile projections, the cone is projected in the form of a triangle.
ka, the width of the base is equal to the diameter of the base. And the height is equal to the height of the cone. The inclined sides of the triangle are projections of the outermost (outline) generatrices of the cone.
Constructing a cone into a rectangle |
||
The isometric view is shown in Fig. 2. |
||
We begin the construction with the location |
||
of the axonometric axes OX, OY, OZ, |
||
holding them at an angle of 1200 to each other. Axis |
||
direct the cone along the OZ axis and set it aside |
||
its height of the cone, obtaining point S. Assume |
||
moving point O beyond the center of the base of the cone, |
||
construct an oval representing the base |
||
cone Then we draw two inclined cables |
||
the nouns from t. S to the oval, which will |
||
extreme (outline) cone-forming |
||
sa. The invisible part of the lower base of the co- |
||
we will draw the nus with a dashed line. |
Constructing points on the surface of a cone in orthogonal and axonometric
sky projections are shown in Fig. 2, 3.
If on the frontal projection of the cone Fig. 2 points A and B are given, then the missing projections
tions of these points can be constructed in two ways.
The first method: using projections of an auxiliary generatrix passing through a given point.
Given: frontal projection of point A – point (a’) located within the visible part of the cone.
Through the vertex of the cone and the given point (a’), we draw a straight line to the base of the cone and get point (e’) - the base of the generatrix s’e’.
H. Let us find the horizontal projection i.e. within the visible part of the circle of the base of the cone by drawing a projecting straight line e’e, and connect the resulting i.e. with the horizontal projection of the vertical
cone tires s.
Since the desired t. A belongs to the image
calling s’e’, then it should lie on its horizontal projection. Therefore, using the communication line, we transfer it to the se line and
we obtain a horizontal projection t. a. Profile projection a” t. A determines
is formed by the intersection of the same generatrix s”e” on the profile projection with the communication lines carrying t.a from the horizontal and frontal
noah projections.
Profile projection a” t. And in this
case, invisible, since it is located behind the projection of the outermost generatrix s”4” and is indicated in parentheses.
Rice. 3 Second method: by constructing projections of a section of a conical surface with a horizontal plane Pv pa-
parallel to the base of the cone and passing through a given point B. Fig. 3. Given: frontal projection of point B – point b’, located within
visible part of the cone.
Through point b’ we draw a straight line Pv parallel to the base of the cone, which
paradise is the frontal projection of the cutting plane P. This line intersects
The axis of the cone lies in point 01’ and the outermost generatrices in points k1’ and k3’. The straight line segment k1’k3’ is the frontal projection of the section of the cone through point b’.
The horizontal projection of this section will be a circle, the radius of which is determined on the frontal projection as the distance 01’k1’ from the co-axis
nous to the extreme generator.
Since point b’ lies in the section plane, using the connection line we transfer it to the horizontal projection of the section within the visible part of the cone.
Profile projection point b” is defined as the intersection of the profile
projection of the section k2”k4” with the communication line transferring the position of point b from the horizontal
zontal projection.
Constructing points on the surface of a cone in axonometry.
We build a cone in rectangular isometry. The construction of the circle of the base of the cone in axonometry repeats the construction of the base of the cylinder. (See section 8.2.1.) Setting aside the height of the cone on the vertical axis, we draw two generatrices - tangent to the base oval.
First way. Rice. 2.
We build the SE generatrix: on the X or Y axis we plot the X or Y coordinates
Y corresponding to i.e. E on the horizontal projection and draw lines through them parallel to the Y or X axis, respectively. Their intersection gives the position of point E at the base of the cone.
Let's connect t. E with the vertex of the cone S and with the center of the base t. 0. Consider the resulting triangle S0E: side 0S is the axis of symmetry of the cone coinciding with the Z axis. Side SE is the generatrix of the cone on which t. A is located. Side 0E is the base of the triangle component with the Z axis angle 900.
The height m. A is taken on the frontal projection perpendicular to the axis
bending the cone to point a’ and putting it in axonometry on the Z axis, that is, on the 0S side.
Through the resulting notch we draw a straight line in the plane of the triangle
parallel to the base of the triangle until it intersects with the SE generatrix. Thus, we transfer the height of position m. A to the surface of the cone
Second way. Rice. 3.
We construct a section of the cone with a plane parallel to the base and passing through the point B. Such a section of the cone is a circle with a radius equal to
segment OK located at a height equal to the height of T.V. In axonometry, this circle is constructed in the form of an ellipse (or an oval replacing it).
Then, on the X and Y axes at the base of the cone, we plot the corresponding
coordinates X and Y t. Taken from the horizontal projection and from the point of their intersection, we restore the perpendicular to the intersection with the section ellipse,
which will determine the position of t.V.
Cone sections.
IN depending on the direction in space of the secant plane passing through the cone, in the section of a right circular cone one can obtain
various flat figures:
A – straight lines (generating) B – hyperbola
B – circle
G – parabola
D - ellipse Conic sections - ellipse, parabola and hyperbola are patterns
natural curves that are constructed from points belonging to the section curve.
A. The section of a cone by a vertical plane passing through its apex is a straight line. Rice. 4.
On the horizontal projection of the cone through point S we draw line Ph at an arbitrary angle to the X and Y axes, which is the horizontal projection of the secant
vertical plane. This line
intersects the circle of the base of the cone at two points a and b, and the segment aob is a horizontal projection of the section of the cone.
Let us mentally discard the left part of the cone from the Ph line and to the right of it we obtain a horizontal projection of the truncated co-
Segments SA and SB - horizontal
projections of the generatrices of the cone along which the cutting plane Ph passes.
We construct generators SA and SB on
frontal projection, transferring points A and B to it and connecting the resulting points a’ and b’ with vertex s’. Triangle a’s’b’ will be the frontal projection of the section
cone, and line s’3’ is the outermost generatrix of the cone.
Similarly, we construct a profile projection of the cone section by moving
points a and b from a horizontal projection onto a profile one and connecting the resulting points a” and b” with the vertex of the cone s”. Triangle a”s”b” is a profile projection of the section of the cone, and line s”2” is the outermost generatrix of the cone.
or X respectively. Their intersection with the line of the base of the cone allows us to obtain points A and B in the axonometry. By connecting them to each other, and each of them
them with the vertex of the cone S, we obtain triangle ABS, which is a section of the cone by the vertical plane P.
B. The section of a cone by a vertical plane that does not pass through its vertex is a hyperbola. Rice. 5.
If the vertical cutting plane P does not pass through the vertex of the cone, then it no longer coincides with the generatrices of its lateral surface, but, on the contrary, intersects
On the horizontal projection of the cone we draw a secant plane Ph at an arbitrary distance from the vertex S and parallel
along the Y axis. In general, the position
The cutting plane relative to the X and Y axes can be anything.
Line Ph intersects the circle of the base of the cone at two points a and b. The segment ab of this line is a horizontal projection
tion of the cone section. We divide the part of the circle to the left of the Ph line into an arbitrary amount
the number of equal parts, in the bottom case by 12 and then each resulting exact
connect the ku on the circle to the vertex of the cone s. These intersection generators
are cut by the cutting plane Ph and we obtain a number of points that belong to the generators and the projection of the section of the cone ab at the same time.
We construct the resulting generators on the frontal projection of the cone
We transfer from the horizontal projection all points on the base of the cone (a, 1, ...,
5, b) and on the frontal projection we obtain points (a’, 1’, ..., 5’, a’) and connect them with the vertex of the cone s’. On the frontal projection through point b’ we draw the cutting plane Pv perpendicular to the base of the cone. Line Pv crosses
all generators and their points of intersection belong to the projection of the section of the cone.
Let's repeat the construction of all the generators on the profile projection of the cone, transferring the points (a, 1, ..., 5, b) from the horizontal projection to it. The resulting points (a”, 1”, …, 5”, b”) are connected to the vertex s”.
We transfer from the frontal projection the points of intersection of the corresponding generators with the cutting plane Pv to the resulting generators. We connect the resulting points with a curved line, which represents a pattern
curve - hyperbola.
Construction of axonometry. Rice. 5.
We build a cone in axonometry, as described above.
Next, from the horizontal projection of the cone, we take coordinates along the X or Y axis for all points a, 1, ..., 5, b and transfer them to the axonometric X or Y axes and find their position on the base of the cone in axonometry. Connecting
them in series with the vertex of the cone S and we obtain a series of generators on the surface of the cone corresponding to the generators on the orthogonal projections.
On each generatrix we find the point of its intersection with the cutting plane P in the same way as described above (see constructing points on the surface of a cone, the first method).
By connecting the points of the pattern curve obtained on the generators, as well as points A and B, we obtain an axonometric projection of the truncated cone.
B Section of a cone by a horizontal plane. Rice. 6.
The cross section of a right circular cone with a horizontal plane parallel to the base is a circle.
If we cut the cone at an arbitrary height h from the base of the cone through point a’
lying on its o’s’ axis with a plane parallel to its base, then on the frontal projection we will see the horizontal line Pv, which is the frontal projection of the cutting plane that forms the section
cones I’, II’, III’, IV’. On profile projection
W view of the cutting plane and the section of the cone are similar and correspond to the line Pw.
On a horizontal projection, a section
cone is a circle in natural
ny value, the radius of the circle of which is projected from the frontal projection as the distance from the axis of the cone at point a’ to point I’, lying on the outermost generatrix 1’s’.
Construction of axonometry. Rice. 6.
We build a cone in axonometry, as described
sano above.
Then on the Z axis we plot the height h of point A from the base of the cone. Through point A we draw lines parallel to the X and Y axes and construct a circle at
axonometry with radius R=a’I’ taken from the frontal projection.
D Section of a cone by an inclined plane parallel to the generatrix. Rice. 7.
We construct three projections of the cone - horizontal, frontal and profile. (see above).
On the frontal projection of the cone, we draw a secant plane Pv parallel to the outline generatrix s’6’ at an arbitrary distance from its origin.
la at the base of the cone through point a’(b’). The segment a’c’ is the frontal projection of the section of the cone.
On the horizontal projection we construct a projection of the base of the cutting plane P through points a, b. The segment ab is the projection of the base of the cone section.
Next, we divide the circumference of the base of the cone into an arbitrary number of parts and connect the resulting points to the vertex of the cone s. We obtain a series of generatrices of the cone, which we successively transfer to the frontal and profile projections. (see point B).
On the frontal projection, the trace of the cutting plane Pv intersects the image
cutting and at the intersection gives a number of points that belong to both the secant plane and the generators of the cone at the same time.
We transfer these points using communication lines to the projections of the generators on the horizon.
zontal and profile projections.
We connect the resulting points with a curved line, which represents
pattern curve - parabola.
Construction of axonometry. Rice. 7.
We construct an axonometric projection of the cone, as described above.
all points (a, b, 1, ..., 6) and transfer them to the axonometric axes X or Y, respectively, thus determining their positions
movement at the base of the cone in axonometry. We connect them in series with the vertex
cone S and we obtain a series of generators on the surface of the cone corresponding to generators on the orthogonal projections.
On each generatrix we find the point of its intersection with the cutting plane P
similar to how it was described above (see constructing points on the surface of a cone).
D. The section of a cone by an inclined plane located at an arbitrary angle to the base of the cone is an ellipse. Rice. 8.
We construct three projections of the cone - horizontal, frontal and pro-
Philine. (see above).
On the frontal projection of the cone, draw a line of the cutting plane Pv at an arbitrary angle to the base of the cone.
On a horizontal projection, we divide the circumference of the base of the cone into an arbitrary number of equal parts (in this case, 12) and obtain
We connect these points to the vertex of the cone S. We obtain a series of generatrices, which, using communication lines, are sequentially transferred to the frontal and profile projections.
On the frontal projection, the cutting plane Pv intersects all generatrices, and the resulting points of their intersection belong simultaneously to the se-
the real plane and the side surface of the cone, being a frontal projection of the desired section.
We transfer these points to the horizontal projection of the cone.
Then we construct a profile projection of the section of the cone (see above), connecting the resulting points of the pattern curve, which is an electric
Construction of the natural size of the section.
Pattern curves (ellipses) on horizontal and profile projections are distorted images of a cross section of a cone.
The true (natural) cross-sectional value is obtained by combining
of the secant plane P with the horizontal plane of projections H. We transfer all points of the cone section on the frontal projection to the X axis using a compass, rotating them around point k". Next, on the horizontal projection, we continue them with connection lines parallel to the Y axis until they intersect with whether-
connection lines taken from the horizontal projection of the corresponding points. Pe-
cutting the horizontal and vertical lines of connection of the corresponding points makes it possible to obtain points belonging to the natural size of the section. By connecting them with a pattern curve, we get an ellipse natural size cone sections.
Construction of axonometry of a truncated cone. Rice. 8.
Constructing an axonometry of a truncated cone is performed by finding points belonging to the section of the cone using any of the methods described above (see above).
Construction of a development of the surface of a truncated cone. Rice. 8.
Let us first construct a development of the lateral surface of the non-truncated
cone We set the position of point S on the sheet and draw an arc from it with a radius equal to the natural value of the length of the generatrix of the cone (for example, s’1’or s’7’). We set the position of point 1 on this arc. We sequentially lay off as many identical segments (chords) from it as the number of parts the circumference of the base of the cone is divided into. The points 1, 2, ..., 12, 1 obtained on the arc are connected to point S. Sector 1S1 is a development of the lateral surface not truncated
fine cone. Having attached to it in the lower part (for example, to point 2) the natural size of the base of the cone in the form of a circle taken from the horizontal projection, we
we obtain a complete development of a non-truncated cone.
To construct a development of the lateral surface of a truncated cone, it is necessary to determine the actual size of all truncated generators. On
of the frontal projection, we transfer all the points of the section to the outline generatrix s’7’ with lines parallel to the base of the cone. Then we transfer each segment of the generatrix from point 7’ to the corresponding point of the section to the corresponding generatrix on the development. By connecting these points on the development, we obtain a curved line corresponding to the section line of the side surface of the
Then apply to the section line on the development (for example, to the generatrix S1)
We construct a natural-size cross-sectional ellipse obtained on the horizontal projection plane H.
Developments of the surface of geometric bodies are drawings
- paper patterns and are used to make the layout of the figure.
A truncated cone is obtained if a smaller cone is cut off from the cone with a plane parallel to the base (Fig. 8.10). A truncated cone has two bases: “lower” - the base of the original cone - and “upper” - the base of the cut off cone. According to the theorem on the section of a cone, the bases of a truncated cone are similar.
The altitude of a truncated cone is the perpendicular drawn from a point of one base to the plane of another. All such perpendiculars are equal (see section 3.5). Height is also called their length, i.e. the distance between the planes of the bases.
The truncated cone of revolution is obtained from the cone of revolution (Fig. 8.11). Therefore, its bases and all its sections parallel to them are circles with centers on the same straight line - on the axis. A truncated cone of revolution is obtained by rotating a rectangular trapezoid around its side perpendicular to the bases, or by rotating
isosceles trapezoid around the axis of symmetry (Fig. 8.12).
Lateral surface of a truncated cone of revolution
This is its part of the lateral surface of the cone of revolution from which it is derived. The surface of a truncated cone of revolution (or its full surface) consists of its bases and its lateral surface.
8.5. Images of cones of revolution and truncated cones of revolution.
A straight circular cone is drawn like this. First, draw an ellipse representing the circle of the base (Fig. 8.13). Then they find the center of the base - point O and draw a vertical segment PO, which depicts the height of the cone. From point P, draw tangent (reference) lines to the ellipse (practically this is done by eye, applying a ruler) and select segments RA and PB of these lines from point P to points of tangency A and B. Please note that segment AB is not the diameter of the base cone, and the triangle ARV is not the axial section of the cone. The axial section of the cone is a triangle APC: segment AC passes through point O. Invisible lines are drawn with strokes; The segment OP is often not drawn, but only mentally outlined in order to depict the top of the cone P directly above the center of the base - point O.
When depicting a truncated cone of revolution, it is convenient to first draw the cone from which the truncated cone is obtained (Fig. 8.14).
8.6. Conic sections. We have already said that the plane intersects the lateral surface of the cylinder of rotation along an ellipse (section 6.4). Also, the section of the lateral surface of a cone of rotation by a plane that does not intersect its base is an ellipse (Fig. 8.15). Therefore, an ellipse is called a conic section.
Conic sections also include other well-known curves - hyperbolas and parabolas. Let us consider an unbounded cone obtained by extending the lateral surface of the cone of revolution (Fig. 8.16). Let us intersect it with a plane a that does not pass through the vertex. If a intersects all the generators of the cone, then in the section, as already said, we obtain an ellipse (Fig. 8.15).
By rotating the OS plane, you can ensure that it intersects all the generatrices of the cone K, except one (to which the OS is parallel). Then in the cross section we get a parabola (Fig. 8.17). Finally, rotating the plane OS further, we will transfer it to such a position that a, intersecting part of the generators of the cone K, does not intersect the infinite number of its other generators and is parallel to two of them (Fig. 8.18). Then in the section of the cone K with the plane a we obtain a curve called a hyperbola (more precisely, one of its “branch”). Thus, a hyperbola, which is the graph of a function, is a special case of a hyperbola - an equilateral hyperbola, just as a circle is a special case of an ellipse.
Any hyperbolas can be obtained from equilateral hyperbolas using projection, in the same way as an ellipse is obtained by parallel projection of a circle.
To obtain both branches of the hyperbola, it is necessary to take a section of a cone that has two “cavities,” that is, a cone formed not by rays, but by straight lines containing the generatrices of the lateral surfaces of the cone of revolution (Fig. 8.19).
Conic sections were studied by ancient Greek geometers, and their theory was one of the peaks of ancient geometry. The most complete study of conic sections in antiquity was carried out by Apollonius of Perga (III century BC).
There are a number important properties, combining ellipses, hyperbolas and parabolas into one class. For example, they exhaust the “non-degenerate”, i.e., curves that are not reducible to a point, line or pair of lines, which are defined on the plane in Cartesian coordinates by equations of the form
Conic sections play an important role in nature: bodies move in gravitational fields in elliptical, parabolic and hyperbolic orbits (remember Kepler's laws). The remarkable properties of conic sections are often used in science and technology, for example, in the manufacture of certain optical instruments or spotlights (the surface of the mirror in the spotlight is obtained by rotating the arc of a parabola around the axis of the parabola). Conical sections can be observed as the boundaries of the shadow of round lampshades (Fig. 8.20).
Depending on the location of the plane section R relative to the axis of a straight circular cone, various cross-sectional figures are obtained, limited by curved lines.
Section of a right circular cone by a frontally projecting plane R is considered in Fig. 182. The base of the cone is located on a plane N. The section figure in this case will be limited to an ellipse.
The frontal projection of the section figure is located on the frontal trace of the plane R(Fig. 182, A).
To construct a horizontal projection of the contour of a sectional figure, the horizontal projection of the base of the cone (circle) is divided, for example, into 12 equal parts. Auxiliary generators are drawn through the division points on the horizontal and frontal projections. First, find the frontal projections of the section points 1′...12", lying on a plane P1. Then, using the communication line, their horizontal projections are found. For example, the horizontal projection of point 2 located on the generatrix s2, is projected onto the horizontal projection of the same generatrix at point 2.
The found horizontal projections of the points of the section contour are connected according to the pattern. The actual appearance of the section figure in this example was found by changing the projection plane. The plane I is replaced by a new plane of projection N 1.
On the frontal plane of projection V section figure - an ellipse is depicted as a straight line 1"7", coinciding with the frontal projection of the cutting plane R. This line 1′7" is the major axis of the ellipse. Minor axis of the ellipse a"b" perpendicular to the major axis 1′ 7" and passes
through its middle. To find the minor axis of the section, through the middle of the major axis 1′7" ellipse draws a horizontal plane N, which will cut the cone along a circle whose diameter will be equal to the minor axis of the ellipse (a 0 b 0).
The construction of a development of the cone surface (Fig. 182, b) begins with drawing a circular arc with a radius equal to length generatrix of a cone from a point S0. The arc length is determined by the angle α:
Where d– diameter of the circle of the base of the cone; l– length of the cone generatrix.
The arc is divided into 12 parts and the resulting points are connected to the vertex s 0. From the vertex, the actual lengths of the generatrix segments from the vertex of the cone to the cutting plane are plotted R.
The actual lengths of these segments are found
as in the example with the pyramid, in a manner near the vertical axis passing through the cone bus. So, for example, to get the actual length of a segment S2, it is necessary to draw a horizontal line from 2′ to the intersection at the point b" with the contour generatrix of the cone, which is its actual length.
The cross-sectional figures and the base of the cone are attached to the development of the conical surface.
Construction of an isometric projection of a truncated cone (Fig. 182, V) start from the base - the ellipse. The isometric projection of any point on the section curve is found from point three coordinates, as shown in Fig. 182, V.
On axis X lay down the dots I...VII, taken from the horizontal projection of the cone. From the obtained points, vertical straight lines are drawn, on which z coordinates taken from the frontal projection are plotted. Through obtained on an inclined axis
ellipse points draw straight lines parallel to the axis y, and the segments 6 0 8 0 and 4 0 10 0 , taken from the actual section view.
The found points are connected according to the pattern. The extreme outlines are drawn tangentially to the contour of the base of the cone and ellipse.
An example of a cross-section of a straight circular cone is shown in Fig. 182, G. The separator cap is a welded structure made of thin sheet steel and consists of two cones.
