They have a large surface area. Which geometric figure has the smallest surface area? Why do fat people sweat?
The ratio of volume to surface area of any physical body. One of the most important engineering techniques.
Imagine a cube with an edge length of 1 meter (1 centimeter, 1 foot, 1 inch or 1 “whatever you want”), then there will be a meter - for simplicity. The volume of this cube is 1 m3. Each side has an area of 1 m2, and the entire surface area of this cube is 6 m2 - there are six sides. The ratio of volume to surface area is 1:6 = 1/6 (now and hereafter - without taking into account dimensions).
Now imagine a cube with a side of 3 m. The volume of this cube is 27 m 3 (3x3x3). Each side has an area of 9 m2, and the entire surface area of this cube is 54 m2. The ratio of volume to surface area is 27:54 = 1/2 = 3/6.
That is, with an increase in linear size by 3 times, the surface area increased by 9 times, but the volume increased by 27 times. The ratio of volume to surface area increased by 3 times.
The table below shows calculations for cubes when doubling the linear size step by step:
Table. Comparison of the dynamics of the surface area and volume of a physical body with increasing linear size.
| Linear size (m) | Surface area (m2) | Volume, m3) |
Ratio of volume to surface area |
|
0,17 |
|||
|
0,33 |
|||
|
0,67 |
|||
|
1,33 |
|||
|
2,67 |
|||
|
5,33 |
|||
|
10,67 |
|||
|
21,33 |
|||
|
42,67 |
|||
|
85,33 |
As the linear size increases, the volume increases much faster than the surface area of the body, since the volume is proportional to the cube of the linear size, and the area is proportional to the square. This fact applies not only to cubic bodies, but also to any other bodies, naturally while maintaining the shape (or proportions, if you prefer).
Drawing. Comparison of the dynamics of the surface area and volume of a physical body with increasing linear size.
Some everyday examples of the importance of the fact in question.1) Heat transfer is proportional to the surface area. Heat capacity is the volume of the body. From this fact it directly follows that a larger building (of the same shape) will take longer to release the heat accumulated during daylight hours (or heat up during the day) and will require less energy per unit of usable area -! The usable area is directly proportional to the internal volume! - for heating (air conditioning).
2) Mass (weight) is proportional to the volume of the support. Soil load - surface area. From this fact it directly follows that for a support of any shape there is a size, starting from which (while maintaining its shape) it will go into any soil.
3) A child has a completely different area/volume ratio than an adult. Therefore, the risks of hypothermia or heatstroke for a child are disproportionately higher (which, of course, is partly compensated by the different speed of metabolic processes in children).
v1=v2. s1>s2. s2. s1. From the wind. From the surface area of the liquid. The larger the surface area of the liquid, the faster evaporation occurs. Water. Water. The wind carries away the vapor molecules. Evaporation occurs faster. Wind.
Slide 11 from the presentation "Evaporation and condensation of liquids". The size of the archive with the presentation is 788 KB.Physics 7th grade
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Their flat edges.
Most often, surface area is determined for the class of piecewise smooth surfaces with a piecewise smooth edge (or without an edge). This is usually done using the following construction. The surface is divided into small parts with piecewise smooth boundaries: in each part, a point is selected at which a tangent plane exists, and the part in question is orthogonally projected onto the tangent plane of the surface at the selected point; the area of the resulting flat projections is summed up; finally, they go to the limit for increasingly smaller partitions (such that the largest of the diameters of the parts of the partition tends to zero). On the specified class of surfaces, this limit always exists, and if the surface is defined parametrically by a piecewise smooth function, where the parameters change in a region on the plane, then the area is expressed by a double integral
where , , , a and are partial derivatives with respect to and . In particular, if a surface is the graph of a -smooth function over a region on the plane, then
Based on these formulas, well-known formulas for the area of a sphere and its parts are derived, methods for calculating the area of surfaces of revolution, etc. are justified.
For two-dimensional piecewise smooth surfaces in Riemannian manifolds, this formula serves as a definition of area, with the role of , , and being played by the components of the metric tensor of the surface itself.
Notes
- An attempt to introduce the concept of the area of curved surfaces as the limit of the areas of inscribed polyhedral surfaces (just as the length of a curve is defined as the limit of inscribed polygonal lines) encounters difficulty. Even for a very simple curved surface, the area of polyhedra inscribed in it with increasingly smaller faces can have different limits depending on the choice of the sequence of polyhedra. This is clearly demonstrated by a well-known example, the so-called Schwartz boot, in which sequences of inscribed polyhedra with different area limits are constructed for the lateral surface of a right circular cylinder.
- It is significant that already in the case of a two-dimensional surface, the area is assigned not to a set of points, but to the mapping of a two-dimensional manifold into space and thus differs from the measure.
see also
Literature
- V. N. Dubrovsky, In search of a definition of surface area. Quantum. 1978. No. 5. P.31-34.
- V. N. Dubrovsky, Surface area according to Minkowski. Quantum. 1979. No. 4. P.33-35.
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See what “Surface area” is in other dictionaries:
surface area- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics: energy in general EN surface areaA ...
Term surface area Term in English surface area, area of interface Synonyms Abbreviations Related terms pores Definition area of the interface, defined as the amount of accessible surface determined by this method... ... Encyclopedic Dictionary of Nanotechnology
surface area- paviršiaus plotas statusas T sritis Standartizacija ir metrologija apibrėžtis Nagrinėjamojo paviršiaus plotas. atitikmenys: engl. surface area vok. Oberflächeninhalt, m rus. surface area, f pranc. aire de surface, f… Penkiakalbis aiškinamasis metrologijos terminų žodynas
surface area- paviršiaus plotas statusas T sritis fizika atitikmenys: engl. surface area vok. Oberflächeninhalt, m rus. surface area, f pranc. aire de surface, f … Fizikos terminų žodynas
Specific surface area- is the total surface area of grains of bulk mineral material or soil, related to its mass (m2/kg) or volume (cm2/cm3). [Handbook of road terms, M. 2005] Term heading: General, fillers Encyclopedia headings: ... ... Encyclopedia of terms, definitions and explanations of building materials
combustion surface area- (in the boiler furnace) [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics: energy in general EN burning surface area ... Technical Translator's Guide
surface area of concentrating mirrors (in a solar power plant)- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN heliostat field ... Technical Translator's Guide
surface area of the collector (solar power plant)- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN collector field ... Technical Translator's Guide
blade surface area- (for example, turbines) [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics: energy in general EN blade area ... Technical Translator's Guide
pore surface area- - Topics oil and gas industry EN pore surface area ... Technical Translator's Guide
Books
- Surface area of forest plants. Essence. Options. Use, Utkin Anatoly Ivanovich, Ermolova Lyudmila Sergeevna, Utkina Irina Anatolyevna. The book combines overview information with materials from its own research. It gives an idea of the surface area of plants, definitions and dimensions of its individual components,...
If the side of a cube is equal to A, That
the volume of the cube will be equal to a 3,
area of one side - a 2, respectively,
area of six sides (i.e. surface area of a cube) - 6a 2. We count:
| A | 1 | 2 | 3 | 4 | 5 | 6 |
| S=6a 2 | 6 | 24 | 54 | 96 | 150 | 216 |
| V=a 3 | 1 | 8 | 27 | 64 | 125 | 216 |
| S/V | 6 | 3 | 2 | 1,5 | 1,2 | 1 |
What do we see? As the size of the cube increases (green line), its surface area (yellow line) gradually increases (from 6 to 216). And the volume of the cube (blue line) also grows (from 1 to 216). Everyone grows, but volume grows faster than surface. You can verify this using the red line, which shows the ratio of surface to volume: per unit of volume at the smallest cube have to six surface units, and the largest has only one.
How can this be assessed? Imagine that each unit of volume is one “person”, and a unit of surface is a window through which a person can breathe. Then
- One person lives in a cube with side 1, and he can breathe through 6 windows;
- 8 people live in a cube with side 2, and they breathe through 24 windows (each gets 3);
- 27 people live in a cube with side 3, and they breathe through 54 windows (each gets 2);
The same for children who do not know how to calculate the area and surface of a cube
Small children! Take the cube in your hands. Do you play with cubes?
No! What are we, little ones? We're playing SonyPlaystation!
Well done children! We took the cubes not to play, but to study biology! Imagine that there is a little man sitting inside the cube, and the sides of the cube are windows through which he can ventilate the room.
Presented! Cool!
The cube has 6 sides, which means that one person has 6 windows and is not stuffy. Now put two cubes together. Now there are 2 people, and there are 10 windows left, that is, 5 for each.
Oops! Here you go!
Now make 4 cubes into a square. There are 4 people, 16 windows, 4 for each. And if you put a second floor, i.e. If you make a super cube 2x2x2, then there will be 8 little people, and 24 windows, 3 for each. Do you feel that it’s getting more and more difficult for the little people to ventilate their rooms?
K – number of cubes, C – number of sides left outside
This topic is complex and obscure. Most of my students never get the hang of it - neither by the ninth grade, nor by the eleventh - but simply remember the rule: the larger the organism, the relatively smaller its surface area, and vice versa. But it’s better not to cram, but to understand, so I strongly recommend that you take your personal dice (which you still play in secret from everyone) and calculate everything yourself. It's worth it: the rule of volume-surface ratio is very often used in our biological farming. Here are a couple of examples.
Doctrine of the Mega Sparrow
Weight birds are volume, multiplied by density, and wing area - this is the surface. From this it becomes clear that as the size of the bird increases, its mass (cubic function) will increase faster than the size of the wings (quadratic function). Slowly growing wings will find it increasingly difficult to lift a rapidly growing mass.
Practical work: take a sparrow and increase its length by 10 times. In this case, the mass of the bird will increase by 1000 times (10 3), and the area of the wings will increase only by 100 times (10 2). We will get a flightless sparrow, the joy of all the predators in the area. To make our mega sparrow fly, we need a second step: increasing the wing area another 10 times. It will be a nice creature!
Why do fat people sweat?
The amount of heat produced by the body depends on the number of cells, i.e. on volume. Heat is transferred to the environment through the surface of the body. Consequently, with increasing body size, heat production (cubic function) increases faster than heat transfer (quadratic function). Therefore, it is difficult for large animals to cool down; they are in danger of overheating (and vice versa, small animals are always at risk of overcooling).
The elephant with its large size has, quite clearly, a very large surface. But relative to volume its surface is very small. In order to get rid of excess heat, the elephant uses its huge ears. They are needed not at all for good hearing (good hearing, for example, in predators - they have small ears), but to increase the surface of the body through which heat transfer occurs.
At this point the children ask: “In India and Africa, isn’t it already so hot there?” Answer: unfortunately, in our cool latitudes, an elephant would not be able to find enough food for itself (and where would it hide during winter?) Mammoths (relatives of the elephant, living in slightly cooler conditions) saved heat: they had normal size ears and fur ( as befits a mammal).
While I was drawing this picture, my wife complained several times that the elephant was a typical alien, just look at him! Indeed, for Russians, an elephant is a completely ordinary animal, even a native one, but this is solely thanks to the talent of Korney Ivanovich Chukovsky: “And the Elephant is a dandy, a hundred-foot merchant’s wife, and the Giraffe is an important count, as tall as a telegraph.” (Chukovsky K.I. “Crocodile”) Residents of other countries, deprived of Chukovsky, perceive the elephant completely differently: “Its knives were like trees, its ears flapped like sails, its long trunk was raised, like a formidable snake ready to pounce, its small eyes inflamed." (Scrombie S. “Delivery of valuable cargo: expert advice”)
If you are interested in the question of which body shape has the smallest total surface area, then you need to keep in mind that the volumes of the bodies being compared must, of course, be the same.
What is needed for the experiment?
To conduct such a research experiment, you will have, in addition to small, simple sculpture lessons that are quite accessible to each of you, to apply knowledge of stereometry. We hope that you find this educational study useful and interesting.
Take a small piece of plasticine, or, if you don’t have it, a piece of well-mashed clay. Make a cube. Try to keep it with equal sides and right angles. Measure the length of its edge and write it down.
Then, form a cylinder from the same cube. The ratio of the size of the bases and the height does not matter. It is important that it is the correct cylinder. Measure the radius of its base and height, and write it down too.
Form a ball from the cylinder. With some effort, you can achieve that you get a real ball. Measure its radius (this is easily done by piercing it with a knitting needle or a straight, stiff wire through its center). After you write down the radius of the ball, if you wish, sculpt other geometric bodies from the ball, for example, a cone, a pyramid, and so on.
Experiment results
And so, you wrote down the sizes of different geometric bodies. They have the most varied shapes, but they have one thing in common - they all have the same volumes. After all, they are all sculpted from one piece of clay or plasticine.
Given the accepted volume of plasticine or clay, for example, one cubic centimeter, you should, after appropriate measurements, have the following approximate data on the total surface area for various figures: ball - 4 square centimeters; cube - 6 centimeters square; cone - 7 centimeters square; cylinder - 8 centimeters square.
Laws of physics
When you blow a soap bubble, it is shaped like a ball.
Have you observed droplets of dew on the leaves of plants in the summer? There are droplets so small that they do not flatten under the influence of their own weight. They look spherical.
Water and other liquids have a thin molecular film on their surface, invisible to the eye. It is elastic near the water. This elastic film always tries to shrink, that is, to take up less space, while forming the smallest surface possible. Have you already seen that the ball has the smallest surface area?
Astronauts in a state of weightlessness can observe how even a portion of water that can fit in a glass melts in the air in the form of a ball. On Earth, under the influence of gravity, water spreads and, in order to preserve it, it is poured into vessels.
But on the surface of an overfilled glass, the bulge formed by water is clearly visible. An invisible molecular film strives to keep water from overflowing. The water film is quite durable. A needle carefully placed on the surface of the water will lie on it, slightly pressed in, forming a small depression.
