The volume of displaced water is equal to the volume of the body. How to calculate buoyancy (buoyant force). Density of the medium and Archimedes' force
Watching the flight of hot air balloons and the movement of ships on the surface of the sea, many people wonder: what makes these vehicles rise into the skies or keeps these vehicles on the surface of the water? The answer to this question is buoyant force. Let's look at it in more detail in the article.
Fluids and static pressure in them
Two aggregate states of a substance are called fluid: gas and liquid. The influence of any tangential force on them causes some layers of matter to shift relative to others, that is, matter begins to flow.
Liquids and gases consist of elementary particles (molecules, atoms) that do not have a specific position in space, as, for example, in solids. They are constantly moving in different directions. In gases this chaotic movement is more intense than in liquids. Thanks to this fact, fluid substances can transmit the pressure exerted on them in all directions equally
Since all directions of movement in space are equal, the total pressure on any elementary volume inside a fluid substance is zero.
The situation changes radically if the substance in question is placed in a gravitational field, for example, in the gravitational field of the Earth. In this case, each layer of liquid or gas has a certain weight with which it presses on the underlying layers. This pressure is called static pressure. It increases in direct proportion to the depth h. Thus, in the case of a liquid with density ρ l, hydrostatic pressure P is determined by the formula:
Here g = 9.81 m/s 2 is the acceleration of free fall near the surface of our planet.
Hydrostatic pressure was felt by every person who dived several meters under water at least once.
Hydrostatic pressure and Archimedes' law
Let's perform the following simple experiment. Let's take a body of regular geometric shape, for example, a cube. Let the length of the side of the cube be a. Let's immerse this cube in water so that its top face is at depth h. What pressure does the water exert on the cube?
To answer the question posed above, it is necessary to consider the amount of hydrostatic pressure that acts on each face of the figure. Obviously, the total pressure acting on all side faces will be equal to zero (pressure on the left side will be compensated by pressure on the right). The hydrostatic pressure on the upper face will be equal to:
This pressure is directed downward. The corresponding force is equal to:
F 1 = P 1 *S = ρ l *g*h*S.
Where S is the area of the square face.
The force associated with hydrostatic pressure acting on the bottom face of the cube will be equal to:
F 2 = ρ l *g*(h+a)*S.
Force F 2 is directed upward. Then the resulting force will also be directed upward. Its value is:
F = F 2 - F 1 = ρ l *g*(h+a)*S - ρ l *g*h*S = ρ l *g*a*S.
Note that the product of the edge length and the area of the face S of the cube is its volume V. This fact allows us to rewrite the formula as follows:
This formula for the buoyant force suggests that the value of F does not depend on the depth of the body’s immersion. Since the volume of the body V coincides with the volume of the liquid V l that it displaced, we can write:
The formula for the buoyancy force F A is usually called the mathematical expression of Archimedes' law. It was first established by an ancient Greek philosopher in the 3rd century BC. Archimedes' law is usually formulated as follows: if a body is immersed in a fluid substance, then it is acted upon by a force directed vertically upward, which is equal to the weight of the substance in question displaced by the body. The buoyant force is also called the Archimedes force or the lifting force.
Forces acting on a solid body immersed in a fluid substance
It is important to know these forces in order to answer the question whether a body will float or sink. In general, there are only two of them:
- gravity or body weight F g ;
- buoyant force F A .
If F g >F A , then we can say with confidence that the body will drown. Conversely, if F g By substituting the formulas for the named forces into the indicated inequalities, one can obtain a mathematical condition for the floating of bodies. It looks like this: Here ρ s is the average density of the body. It is not difficult to demonstrate the effect of the above condition in practice. It is enough to take two metal cubes, one of which is solid and the other is hollow. If you throw them into water, the first one will drown, and the second one will float on the surface of the water. All vehicles that move on the surface of water or under water use Archimedes' principle. Thus, the displacement of ships is calculated based on knowledge of the maximum buoyancy force. Submarines, by changing their average density with the help of special ballast chambers, can float or submerge. A striking example of changes in average body density is the use of life jackets by humans. They significantly increase the total volume and at the same time practically do not change the person’s weight. The rise of a balloon or children's balloons inflated with helium in the sky is a clear example of the action of the Archimedean buoyant force. Its appearance is associated with the difference between the density of hot air or gas and cold air. A hollow ball is completely immersed in water. The radius of the ball is 10 cm. It is necessary to calculate the buoyant force of water. To solve this problem, you do not need to know what material the ball is made of. You just need to find its volume. The latter is calculated by the formula: Then the expression for determining the Archimedean force of water will be written as: F A = 4/3*pi*r 3 *ρ l *g . Substituting the radius of the ball and the density of water (1000 kg/m3), we find that the buoyant force is 41.1 N. There are two bodies. The volume of the first is 200 cm 3, and the second is 170 cm 3. The first body was immersed in pure ethyl alcohol, and the second in water. It is necessary to determine whether the buoyancy forces acting on these bodies are the same. The corresponding Archimedean forces depend on the volume of the body and the density of the liquid. For water, the density is 1000 kg/m3, for ethyl alcohol - 789 kg/m3. Let's calculate the buoyancy force in each fluid using these data: for water: F A = 1000*170*10 -6 *9.81 ≈ 1.67 N; for alcohol: F A = 789*200*10 -6 *9.81 ≈ 1.55 N. Thus, in water the Archimedean force is 0.12 N greater than in alcohol. Instructions Archimedean force arises due to the difference in water pressure at the level of the upper and lower sections of the body. A column of water of height h1 presses on the upper part with a force equal to the weight of this . The lower part is acted upon by a force equal to the weight of the column of height h2. This height is determined by the addition of h1 and the height of the body itself. According to Pascal's law, pressure in a liquid or gas is distributed evenly in all directions. Including upwards. Obviously, the force acting upward is greater than the force acting downward. But, it should be noted that only the effect of the liquid column is taken into account. The buoyancy force does not depend on the body's own weight. Neither the material from which the body is made, nor its other qualities, except for its dimensions, are used in calculations. The calculation of the Archimedean force is based only on the density of the liquid and the geometric dimensions of the immersed part. There are two ways, the Archimedean force acting on a body immersed in a liquid. The first consists of measuring the volume of a body and calculating the weight of a liquid occupying a similar volume. To do this, it is necessary that the body has the correct geometric shape, that is, it is a cube, parallelepiped, sphere, hemisphere, cone. It is very difficult to calculate the volume of a solid body of a more complex shape, so to determine the Archimedes force in this case there is a more practical method No. 2. But more on that later. Having determined the volume of the immersed body, we multiply it by the density of the liquid and find the magnitude of the buoyancy force acting on this body in a homogeneous medium of a given density and on the acceleration of free fall g (9.8 m/s2). The formula for determining the Archimedes force looks like this: The second method is based on measuring the volume of displaced liquid. It most closely corresponds to the experience that led Archimedes to the discovery of his law. This method is also very convenient when calculating the Archimedean force for partial immersion of a body. To obtain the necessary data, the body under study is suspended on a thread and slowly lowered into the liquid. It is enough to measure the level of liquid in the vessel before and after immersion of the body, multiply the difference in levels by the surface area and find the volume of displaced liquid. As in the first case, we multiply this volume by the density of the liquid and g. The resulting value is the Archimedes force. For the unit of force to be the Newton, the volume must be measured in m3 and the density in kg/m3. ARCHIMEDES' LAW– the law of statics of liquids and gases, according to which a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid in the volume of the body. The fact that a certain force acts on a body immersed in water is well known to everyone: heavy bodies seem to become lighter - for example, our own body when immersed in a bath. When swimming in a river or in the sea, you can easily lift and move very heavy stones along the bottom - ones that we cannot lift on land; the same phenomenon is observed when, for some reason, a whale is washed up on the shore - the animal cannot move outside the aquatic environment - its weight exceeds the capabilities of its muscular system. At the same time, lightweight bodies resist immersion in water: sinking a ball the size of a small watermelon requires both strength and dexterity; It will most likely not be possible to immerse a ball with a diameter of half a meter. It is intuitively clear that the answer to the question - why a body floats (and another sinks) is closely related to the effect of the liquid on the body immersed in it; one cannot be satisfied with the answer that light bodies float and heavy ones sink: a steel plate, of course, will sink in water, but if you make a box out of it, then it can float; however, her weight did not change. To understand the nature of the force acting on a submerged body from the side of a liquid, it is enough to consider a simple example (Fig. 1). Cube with an edge a immersed in water, and both the water and the cube are motionless. It is known that the pressure in a heavy liquid increases in proportion to depth - it is obvious that a higher column of liquid presses more strongly on the base. It is much less obvious (or not at all obvious) that this pressure acts not only downwards, but also sideways and upwards with the same intensity - this is Pascal's law. If we consider the forces acting on the cube (Fig. 1), then due to the obvious symmetry, the forces acting on the opposite side faces are equal and oppositely directed - they try to compress the cube, but cannot affect its balance or movement. There remain forces acting on the upper and lower faces. Let h– depth of immersion of the upper face, r– fluid density, g– acceleration of gravity; then the pressure on the upper face is equal to r· g · h = p 1 and on the bottom r· g(h+a)= p 2 The pressure force is equal to the pressure multiplied by the area, i.e. F 1 = p 1 · a\up122, F 2 = p 2 · a\up122 , where a- cube edge, and strength F 1 is directed downwards and the force F 2 – up. Thus, the action of the liquid on the cube is reduced to two forces - F 1 and F 2 and is determined by their difference, which is the buoyancy force: F 2 – F 1 =r· g· ( h+a)a\up122 – r gha· a 2 = pga 2 The force is buoyant, since the lower edge is naturally located below the upper one and the force acting upward is greater than the force acting downward. Magnitude F 2 – F 1 = pga 3 is equal to the volume of the body (cube) a 3 multiplied by the weight of one cubic centimeter of liquid (if we take 1 cm as a unit of length). In other words, the buoyant force, which is often called the Archimedean force, is equal to the weight of the liquid in the volume of the body and is directed upward. This law was established by the ancient Greek scientist Archimedes, one of the greatest scientists on Earth. If a body of arbitrary shape (Fig. 2) occupies a volume inside the liquid V, then the effect of a liquid on a body is completely determined by the pressure distributed over the surface of the body, and we note that this pressure is completely independent of the material of the body - (“the liquid doesn’t care what to press on”). To determine the resulting pressure force on the surface of the body, you need to mentally remove from the volume V given body and fill (mentally) this volume with the same liquid. On the one hand, there is a vessel with a liquid at rest, on the other hand, inside the volume V- a body consisting of a given liquid, and this body is in equilibrium under the influence of its own weight (the liquid is heavy) and the pressure of the liquid on the surface of the volume V. Since the weight of liquid in the volume of a body is equal to pgV and is balanced by the resultant pressure forces, then its value is equal to the weight of the liquid in the volume V, i.e. pgV. Having mentally made the reverse replacement - placing it in volume V given body and noting that this replacement will not affect the distribution of pressure forces on the surface of the volume V, we can conclude: a body immersed in a heavy liquid at rest is acted upon by an upward force (Archimedean force), equal to the weight of the liquid in the volume of the given body. Similarly, it can be shown that if a body is partially immersed in a liquid, then the Archimedean force is equal to the weight of the liquid in the volume of the immersed part of the body. If in this case the Archimedean force is equal to the weight, then the body floats on the surface of the liquid. Obviously, if, during complete immersion, the Archimedean force is less than the weight of the body, then it will drown. Archimedes introduced the concept of "specific gravity" g, i.e. weight per unit volume of a substance: g =
pg; if we assume that for water g= 1, then a solid body of matter for which g> 1 will drown, and when g < 1
будет плавать на поверхности; при g= 1 a body can float (hover) inside a liquid. In conclusion, we note that Archimedes' law describes the behavior of balloons in the air (at rest at low speeds). Vladimir Kuznetsov The phenomenon in which a body, due to the action of a buoyant force directed upward, does not sink when immersed in liquids or gases is called buoyancy. The buoyant force is a balancing force that acts opposite to the force of gravity. It is worth noting here that not only liquid can act, but also gas, and even metal! Floating objects on water, under water, in the air: ships, submarines and balloons make their movement due to the balance of gravity and buoyancy. Traditionally, we will try to explain the nature of the buoyancy of objects in simple language without diving into dense formulas. Any body on Earth falls down under the influence of gravity and the resulting corresponding downward gravity force. But, for sure, we all noticed that when an object is immersed in water, it becomes lighter. This means that when objects are immersed in liquid, some other buoyant force begins to act, directed oppositely to the force of gravity upward.
The buoyant force that counteracts the force of gravity is called Archimedean, in honor of the famous Greek scientist Archimedes, who lived in the 3rd century BC. According to Archimedes' law, any body immersed in a liquid is subject to a buoyant force, the magnitude of which is equivalent to the weight of the body. The mass of the pear does not change when immersed in liquid, but its weight decreases by the value of the buoyancy force from the water. Next, imagine a cube of concrete whose mass is 3000 kg or 3 tons. From the initial physics course we get the weight (mass of the cube multiplied by the acceleration due to gravity h=9.8) of a concrete cube. Roughly, this is the amount of mass multiplied by 10. So the weight of the concrete cube is P = 30000N (Newton). So, when a given volume of concrete is immersed in water, water weighing 1000 kg or 1 ton is pushed out. The buoyant force opposing the weight of a concrete cube is 10,000 N. It is by this amount that the weight of the cube ultimately decreases when completely immersed in water. 30000N-10000N=20000N. This is where the whole effect of weight reduction lies, which is what we feel under water. As we see from our experiment, during the dive the body lost one third of its weight. Bodies immersed in any fluid medium experience pressure directed from all sides of this very medium and the magnitude of which increases as they are immersed. Accordingly, the pressure that the medium exerts on a body with a certain height difference will be maximum at the bottom point (plane) of the body, and minimum at the top. The direction of pressure forces on the upper and lower planes are respectively opposite. The resultant of these two opposite forces is the buoyant force. Returning to our concrete cube, submerged, for example, to a depth of 1 m, it is subject to pressure forces from water on six sides. Since the sides of the cube are at the same depth, the resulting forces balance the forces directed towards each other. What about the pressure forces on the lower and upper plane? Here, on the upper plane of the cube, the magnitude of the force directed downward is equal to 10,000 N, and on the lower plane, the force of 20,000 N is directed upward. The buoyancy force is equal to the difference in forces acting between the bottom and top - 10000N. A floating body, such as a ship, pushes a volume of water out of the space it occupies instead of water. The weight of the ejected volume of water is equal to the weight of the vessel. The buoyancy of a body is explained by water pressure. Many of us wonder why the same bodies with the same volume float, while others do not, that is, drown. The above-mentioned concrete cube will definitely sink in water, but the wood cube will remain afloat. Let's take a wooden cube weighing 500 kg. Its weight is 5000N. According to Archimedes' law, a wooden cube will displace the same volume of water as a cube of concrete - 1 m3. The mass of displaced water will also remain unchanged - 1000 kg or 1 ton. It turns out that the buoyancy of a wooden cube is due to the fact that the resulting buoyant force is greater than the weight of the wood. It's simple, isn't it? But what about the fact that a floating body, in particular our wooden cube, is not completely, but only partially immersed in water? And at the same time, water is pushed out by 1000 kg, or less. This effect speaks precisely of the balancing of the buoyant force and body weight. Underwater is that part of the body that is capable of creating a buoyancy force equal to the weight of the body above the water. This balance of forces answers the question why the body floats. If you start adding weight to a floating body from above, then the volume of the displaced liquid increases with its buoyant force and grows directly exactly by the amount of added weight. As a result, one can observe an increasing immersion of the body under water. The submarine consists of a waterproof hull, under the skin of which tanks with ballast are located. When the tanks are filled with water, the boat sinks. Underwater, the boat is suspended - it neither sinks nor floats. When it is necessary to ascend, air is pumped into the ballast tanks, which displaces the water outward. Markings on the side of the ship show how much cargo it can take for safe navigation. The degree of immersion of a vessel depends, in particular, on the density of the water in which it is located. So, we found out that depending on whether the buoyancy force is greater or less than the weight of the body, the property of its buoyancy is determined. If the buoyancy force is greater, the body floats; if it is less, it sinks. The direct connection between whether a body will float is the ratio of the density of the medium to the density of the body immersed in this very medium (liquid or gas). Let's remember the physics course - the density of a body is the ratio of its mass to volume. In our experiments, the following substances with the corresponding densities were used: concrete - 3000 kg/m3, wood - 500 kg/m3 and water 1000 kg/m3. But what about ships, which for the most part are made of metals whose density significantly exceeds the density of water? And as a result, the calculated part of the density includes this very volume of air in the hollow part. As a result, the resulting part of the buoyant force is greater than the body weight. A hydrometer is a device for measuring the density of a liquid. The greater the density of the liquid, the greater the buoyancy force, and the higher the body of the device floats. All of the above applies not only to liquid media, but also to gas media. Let's take everyone's favorite balloons. They also swim, but only in the air. The air heated by the burner inside the ball has a lower density than the surrounding air at a lower temperature. As a result, the balloon lifts off the ground. What about the magic of instantly rising balloons filled with a gas called helium? Here again, it’s all about the difference in the density of helium and air gases. The density of helium is less than air, so tenches easily fly into the air for festive events. : a body immersed in a liquid (or gas) is subject to a buoyancy force equal to the weight of the liquid (or gas) displaced by this body. The force is called by the power of Archimedes: where is the density of the liquid (gas), is the acceleration of gravity, and is the volume of the submerged body (or the part of the volume of the body located below the surface). If a body floats on the surface or moves uniformly up or down, then the buoyant force (also called the Archimedean force) is equal in magnitude (and opposite in direction) to the force of gravity acting on the volume of liquid (gas) displaced by the body, and is applied to the center of gravity of this volume . A body floats if the Archimedes force balances the force of gravity of the body. It should be noted that the body must be completely surrounded by liquid (or intersect with the surface of the liquid). So, for example, Archimedes' law cannot be applied to a cube that lies at the bottom of a tank, hermetically touching the bottom. As for a body that is in a gas, for example in air, to find the lifting force it is necessary to replace the density of the liquid with the density of the gas. For example, a helium balloon flies upward due to the fact that the density of helium is less than the density of air. Archimedes' law can be explained using the difference in hydrostatic pressure using the example of a rectangular body. Where P A, P B- pressure at points A And B, ρ - fluid density, h- level difference between points A And B, S- horizontal cross-sectional area of the body, V- volume of the immersed part of the body. In theoretical physics, Archimedes' law is also used in integral form: where is the surface area, is the pressure at an arbitrary point, integration is carried out over the entire surface of the body. In the absence of a gravitational field, that is, in a state of weightlessness, Archimedes' law does not work. Astronauts are quite familiar with this phenomenon. In particular, in zero gravity there is no phenomenon of (natural) convection, therefore, for example, air cooling and ventilation of the living compartments of spacecraft is carried out forcibly by fans. A certain analogue of Archimedes' law is also valid in any field of forces that act differently on a body and on a liquid (gas), or in a non-uniform field. For example, this refers to the field of inertial forces (for example, centrifugal force) - centrifugation is based on this. An example for a field of a non-mechanical nature: a conducting body is displaced from a region of a magnetic field of higher intensity to a region of lower intensity. There is hydrostatic pressure of the fluid at depth. In this case, we consider the fluid pressure and the gravitational field strength to be constant values, and - a parameter. Let's take a body of arbitrary shape that has a non-zero volume. Let's introduce a right-handed orthonormal coordinate system, and choose the direction of the z axis to coincide with the direction of the vector. We set zero along the z axis on the surface of the liquid. Let us select an elementary area on the surface of the body. It will be acted upon by the fluid pressure force directed into the body. To get the force that will act on the body, take the integral over the surface: When moving from the surface integral to the volume integral, we use the generalized Ostrogradsky-Gauss theorem. We find that the modulus of the Archimedes force is equal to , and it is directed in the direction opposite to the direction of the gravitational field strength vector. The behavior of a body located in a liquid or gas depends on the relationship between the modules of gravity and the Archimedes force, which act on this body. The following three cases are possible: Another formulation (where is the density of the body, is the density of the medium in which it is immersed): Wikimedia Foundation. 2010. ARCHIMEDES' LAW, ARCHIMEDES concluded that a body immersed in a liquid is pushed out with a force equal to the weight of the displaced liquid. It is said that he allegedly formulated this law by immersing himself in a bathtub and watching the water flow out. According to… … Scientific and technical encyclopedic dictionary ARCHIMEDES' LAW- the law of hydro and aerostatics, according to which any body immersed in a liquid or gas is acted upon by a buoyant force (Archimedean force), equal to the weight of the liquid (gas) displaced by the body, directed vertically upward and applied to the center... ... Big Polytechnic Encyclopedia Archimedes' law- Archimedo dėsnis statusas T sritis Standartizacija ir metrologija apibrėžtis Skysčių ir dujų statikos dėsnis: kūną, panardintą į skystį ar dujas, veikia išstumiamoji jėga F, lygi kūno išstumto skysčio ar dujų sunkiui; jos veikimo taškas –… … Penkiakalbis aiškinamasis metrologijos terminų žodynas Archimedes' law- Archimedo dėsnis statusas T sritis fizika atitikmenys: engl. Archimedes law; Archimedes principle vok. Archimedisches Gesetz, n; Archimedisches Prinzip, n rus. Archimedes principle, m; Archimedes' law, m pranc. principe d'Archimède, m; théorème… … Fizikos terminų žodynas ARCHIMEDES' LAW: any body immersed in a liquid is acted upon by a buoyant force directed upward and equal to the weight of the liquid displaced by it. Archimedes' law is also true for gases... encyclopedic Dictionary Archimedes' law- Archimedes’ law Archimed’s law *Archimedisches Prinzip – on a body that is entangled in the middle, a force is vertically directed upward, which is equal to the force of gravity of the body, which is equal to the volume of the encumbered body. If the gravitational force of the body G is greater... ... Girnichy encyclopedic dictionary This term has other meanings, see Law (meanings). A physical law is an empirically established and expressed in strict verbal and/or mathematical formulation, a stable connection between repeating phenomena, processes and... ... Wikipedia Archimedes' law- Archimedes' law: F buoyant force; P is the force of gravity acting on the body. ARCHIMEDES LAW: any body immersed in a liquid is acted upon by a buoyant force directed upward, equal to the weight of the liquid displaced by it and applied to the center... ... Illustrated Encyclopedic Dictionary
Application of buoyant force in practice
The problem of calculating the Archimedean force in water
Archimedean forces comparison problem
F=ρgV
ρ is the specific density of the liquid;
g - free fall acceleration;
V is the volume of displaced liquid.
Like any force, it is measured in Newtons (N).
Archimedes' law and the floating of bodies
Immersion of the body in water
Nature of buoyant force
Why does the body float?
Dependence of buoyancy on body density
Generalizations
Derivation of Archimedes' law for a body of arbitrary shape
Condition of floating bodies
see also
Notes
Links
See what "Archimedes' Law" is in other dictionaries:
