A body immersed in a liquid will sink. Formulation and explanation. Elements of continuum mechanics
STABILITY OF SUBMARINES
ELEMENTS OF CONTINUOUS MEDIA MECHANICS
A continuous medium is considered to be characterized by a uniform distribution of matter - i.e. medium with the same density. These are liquids and gases.
Therefore, in this section, we will consider the main laws that hold in these environments.
There are a number of factors involved in whether an object will float or submerge in liquid. Objects placed in liquid displace their volume of liquid. If the density of the object is less than the density of the liquid, the object will float. Density is also related to the specific gravity of an object. The weight of an object in a fluid is less than its weight outside the fluid.
Questions you may have about swimming.
- What is movement?
- What is density and specific gravity?
- How does everything float?
Topic 1.9 STATIC LIQUIDS
LAW OF ARCHIMEDES
Consider Figure 1, which shows a vessel with water in which a certain body of arbitrary shape is immersed. During the dive, it is clear that
Rice. one the water level is rising, i.e. body
releases a certain amount
Water. It is also clear that this number
Why don't all bodies sink
Gravity pulls a denser object down. The smaller dense object is pushed upwards. Swimming balance. The density of a metal vessel is less than water. When a solid object is placed in a liquid, it displaces its volume in the liquid. In other words, it replaces a volume of liquid equal to its own volume.
For example, if you placed 1 cubic meter in a full container of water, 1 cubic meter of water will flow down the sides of the container. Of course, if you place an object in a container that won "overflow", the liquid level will rise according to the volume added.
Vt of water is equal to the volume of the submerged body.
FROM This body is affected by data
BUT conditions, there are 2 forces:
one). Gravity F t, which is directed
vertically down and attached to
Point C, which is called the center
body gravity.
2). A certain force that is directed vertically upwards and is applied to point A, which is called the center of buoyancy of the body. This force was discovered and calculated by the ancient Greek mathematician Archimedes, and therefore named after him the Archimedean force. Archimedean force has the formula:
The displacement principle is a factor in why some objects float. It is also handy for finding the volume of irregularly shaped objects. The amount or amount of space that an object occupies is related to its density. The density of a material is how closely its matter is packed and represented as its mass per unit volume. Specific gravity is a comparison of the density of a material with the density of water.
The density of a material is a combination of the atomic weight of its atoms and how tightly packed the atoms or molecules are. Various elements and materials have different natural densities. Water is denser than engine oil. Although 100 pounds of feathers can take up much more space than 100 pounds of steel, they both still weigh 100 pounds. Steel is heavier in size due to the fact that it is a denser material.
F Α = ρ x gV t,
where F Α is the force of Archimedes, ρ w is the density of the liquid in which the body is immersed, g \u003d 9.8 m / s 2 V t is the volume of the body.
If the immersed volume changes, then the buoyant force acting on the body will also change. Then the last formula takes the differential form:
dF Α = ρ x gdV t,
But what is the center of buoyancy? This is the point that is the center of gravity of the displaced fluid. Those. if there was liquid in place of the body, then this volume of liquid would have its own center of gravity - in this place is the center of buoyancy of the body. The center of buoyancy is located in the geometric center of the immersed volume of the body, while the center of gravity is not always located in the geometric center of the body, because density inside solid body may be unevenly distributed.
Comparing the density of a solid to that of a liquid will determine if an object will float in the liquid. The density of a material is its mass divided by its volume. So a material like feathers takes up much more space than a denser material like steel for the same mass or weight.
Density of water in metric system is 1 by the definition of units: 1 cubic centimeter of water weighs 1 gram. Because density is related to the mass of an object, the density of a given volume of lead will be the same on both Earth and the Moon, although lead will weigh less on the Moon due to lower gravity.
So, law of Archimedes is formulated as follows: A body immersed in a liquid is subjected to a force equal to the weight of the displaced liquid and applied to the body's center of buoyancy.
From the ratio of forces acting on a body immersed in a liquid, it is easy to obtain float condition:
If a F t > F A, then the body sinks;
AT English system The pound is indeed a unit of weight on Earth, not mass. But we can still use pounds to get an idea of density. If you know the natural density of a pure element and you know the weight and volume of an object, you can calculate its density and thus determine whether it is a pure element.
Specific gravity is often used instead of density when comparing materials to water. The ratio of the density of an object compared to the density of water is defined as the specific gravity of the material. Specific gravity is often used to compare liquids mixed with water. It is also used to determine the amount of acid in battery your vehicle.
If a F t< F А , then the body floats;
If a F t \u003d F A, then the body floats at the level where these forces are equalized.
Rice. 2 Possible destinations
From fig. 1 shows that the Archimedean force is directed vertically upwards. And if the body is washed by liquid from all sides, then F A has just such a direction. But, in case when there is no water layer between the body and the bottom ( those. the body lies firmly on the bottom - see Fig. 2 ), then the Archimedean force is directed perpendicular to the bottom surface .
If you push a piece of wood underwater, the force will pull it to the top surface where it will float. Likewise, when you hold a balloon filled with helium, you may feel a force pulling the balloon up. The force that pushes a submerged object upward is the result of fluid pressure from gravity and the difference in density between the object and the fluid.
Wording and Explanations
At any depth in a fluid, the pressure in all directions is proportional to that depth. Thus, the water pressure at 10 meters is twice the pressure at 5 meters. An object under water will have a downward pressure on its top proportional to the depth of its top, and it will have an upward pressure on its bottom proportional to the depth of the bottom.
Figure 3 shows an experiment confirming the above
Rice. 3 the mentioned statement. Bottom of a glass vessel
covered with a thin layer of paraffin. They put on it
a piece of paraffin with a smooth base and carefully
pour water into a vessel. A piece of paraffin
floats to the surface of the water, although the density
If the object weighed more than an equal volume of liquid—in other words, its density was greater—then the downward pressure at its bottom would be the weight of the liquid above plus the weight of the object. This force would be greater than the upward force due to the weight of the water alone. Since the force of the downward movement is greater than the force of the upward movement, the object descends.
An example is a concrete block dropped into water. The block's density, or specific gravity, is greater than that of water, so the pressure at the bottom of the block from its weight will be greater than the water's upward pressure at the point, and it will sink. If an object weighs less than the fluid it is displacing and sinks into the fluid, the upward pressure will be greater than the downward pull and it will float.
its less than the density of water. Slightly tilting the vessel,
you can make a piece of paraffin move along
bottom, but it does not float. But if you tilt a piece of paraffin so that water penetrates under its bottom surface, then the buoyant force will cause the paraffin to float vertically upwards. It is known that a submarine lying on soft ground sometimes cannot break away from it, even when its ballast compartments are completely free of water. This is also due to the fact that water cannot penetrate under the hull of the boat, which is firmly attached to the ground.
Wood is less dense than water. When it is placed in water, the water it displaces is equal to its weight, not its volume. Thus, the heavier water is pulled to the bottom by gravity, pushing the wood towards the top. When an object floats in a fluid, the weight of the fluid being moved is equal to the weight of the object. In other words, the water weight of a wooden block that floats halfway above the surface of the water is equal to the weight of the block. If you add weight to the top of the block, the same weight of water will be displaced.
Boats or ships made of steel are hollow. The total weight is less than the water it displaces and thus the ship will float. Different liquids may have different densities. For example, because pint oil is lighter in weight than the same volume of water, it is less dense than water. Different gases can also have different densities. For example, helium is lighter than air because it is less dense.
STABILITY OF VESSELS
Stability is the ability of a ship to return to its equilibrium position after it has been taken out of it. .
Let's remember what equilibrium positions are. It's believed that a body is in equilibrium if the action of all forces applied to it is compensated . Those. in equilibrium, the body must be in a state of rest or uniform and rectilinear motion. There are three types of balance:
When liquids and gases are buoyant
The ship will displace a volume of water equal to the weight of the ship. This is why a loaded cargo ship will sit lower in the water than an unloaded one. The difference will be equal to the weight of the load. When you add liquids of different densities, the higher density liquid tends to settle to the bottom of the container. A denser liquid more pressure due to gravity, thereby pushing the lighter material out of its way. This is why oil will float on top of the water and why air bubbles also rise up into the water.
sustainable, unstable and indifferent. Figure 4 clearly shows the difference between them. In Fig. 4(a), a small ball is placed inside
Rice. 4 Types of equilibria concave hemisphere. If you are his
a). b). in). give yourself, he
stop at the lowest point
An interesting observation is that the bubbles that start at the bottom of the lake get bigger as they get closer to the surface. The reason is that the water pressure is less when they rise. This is also true for a balloon filled with air placed at different depths in the water.
Just as an air-filled balloon is much less dense than the water it displaces, and the weight or pressure of the water pushes the balloon up to the top of the water, a helium-filled balloon is lighter than its volume of air and therefore rises. up into the atmosphere.
hemisphere, which corresponds
stable unstable indifferent equilibrium position. And if
the ball will hang out of this position, it will return there anyway, and without outside help. Such an equilibrium is called stable. And Fig. 4(b) also shows equilibrium: the ball is located at the highest point of the hemisphere, but as soon as you touch it, it will leave the equilibrium position and will not return to it by itself. Such an equilibrium is unstable.
If you put an object that didn't float in a liquid and didn't weigh that object when it was immersed in that liquid, it would weigh its weight in air minus the weight of the liquid it moved. In other words, if you put a brick in a bucket of water filled to the brim and weighing the brick in the water, the measured weight will be the weight of the brick in air minus the weight of the water that overflowed from the bucket.
Submerged objects displace their volume in the liquid. The density of an object is its mass per unit volume. Objects float when their density is less than liquid. This is because the downward pressure of the object's mass is less than the upward pressure of the fluid at that depth.
Let's take a look at how stability works. For ships and ships, the question of the stability of their equilibrium during navigation (the stability of ships) is extremely important. It is known that if the cargo is not properly distributed on the ship, it can capsize. The question of stability is a question of navigational safety.
In order for a ship (or vessel) to be in a state of stable equilibrium, its center of gravity C and its center of buoyancy A must be located on the same vertical.
The state of equilibrium of a solid body partially or completely immersed in a liquid or gas is called floating. The theory of navigation is connected mainly with the determination of the positions of equilibrium of a body immersed in a fluid, and with the establishment of conditions for the stability of the equilibrium. Archimedes' principle gives the simplest condition for bodies to float.
When the couple seeks to return the body to a position of equilibrium, the equilibrium is stable; otherwise it is unstable. The stability of the equilibrium can be assessed by the position of the metacenter. You know for sure that the stone sinks into the water, but the piece of wood floats. But what about a coin, a piece of Styrofoam, a raw chicken egg, a tomato, a piece of potato, or a crumpled ball?
STABILITY OF SUBMARINES
Consider the stability of the equilibrium of a body under water - for example, a submarine.
Rice. 5 Stability of submarines
Figure 5(a) is a schematic representation of a submarine (in cross section).
Two forces act on it: Archimedean force and gravity, which balance each other. Note that the center of buoyancy A is located above the center of gravity C. If the center of buoyancy were below the center of gravity, then the balance would be unstable.
What qualities may be important for swimming?
- First put your guess in the table.
- Then check with your group if your guess is correct and write down the answer!
What does it matter if the object is floating in water? You have now discovered that mass plays a role in whether a body floats or sinks into water. The same cubes have the same shape, but a different mass. You can swim or sink to the bottom. Now we have looked at the shape and mass of objects, and some have experimented. Why does one body float and the other not?
When heeling (Fig. 5 (b)) these two forces still act on the submarine in the same directions, but this time points C and A do not lie on the same vertical, and the axes of forces lie on two parallel lines. and form a pair of forces that will return the boat to its original position. The center of buoyancy A in this case plays the role of a suspension point.
How does the Archimedean Principle work? Every body has a certain shape and mass. Mass is measured in kilograms. The body presses its mass on the surface of the water. Under the influence of gravity, it sinks into the water. When immersed, water particles are pressed down and sideways. The resulting force is called the physicist's weight. Particles of the medium develop counteraction.
Whether an object floats on water or not depends on the relationship between buoyancy and weight. He made similar attempts and formulated them in law. These are: the buoyancy of a body in a medium is as great as the mass of the body displaced by the medium. By the way, the discovery of the famous Greek also works with gases. Everything flies that has a lower density than air. All things that have the same density as the air float, and objects with a higher density, remain on the ground. This is measurable by filling a glass with water and placing it in a bowl.
STABILITY OF VESSELS
The conditions for the stability of the equilibrium of a body floating on the surface of a liquid will be completely different, since when a body (for example, a ship) is tilted, the shape of the displaced volume changes, and, consequently, the position of the center of buoyancy relative to the ship (see Fig. 6). for example, when heeling to the right (Fig. 6 (b)) most of the displaced water will be located to the right of the vessel's own axis, and, consequently, the center of buoyancy will shift to the same side.
Rice. 6 Stability of ships
a). b).
As can be seen in the figure, here the issue of stability depends on the relative position of the center of buoyancy A and the center of gravity C when heeling.
Point M is the point of intersection of the diametrical plane of the vessel with the axis of the Archimedes force - is called the metacenter. If the metacenter lies above the center of gravity, then the pair of forces F A and G turns the ship back to its original position, hence the equilibrium is stable. Here, the role of the suspension point is played by the metacenter M. Note that the position of the metacenter changes with a change in the ship's roll angle.
The distance of the metacenter M from the center of gravity C is called the metacentric height. Using this concept, one can formulate
SHIP STABILITY CONDITION:
Different objects in a liquid behave differently. Some sink, others stay on the surface and float. Why this happens, explains the law of Archimedes, discovered by him under very unusual circumstances and which became the basic law of hydrostatics.
How did Archimedes discover his law?
The legend tells us that Archimedes discovered his law by accident. And this discovery was preceded by the following event.
King Hieron of Syracuse, who ruled in 270-215. BC, suspected his jeweler that he mixed a certain amount of silver into the gold crown ordered to him. To dispel doubts, he asked Archimedes to confirm or refute his suspicions. As a true scientist, Archimedes was fascinated by this task. To solve it, it was necessary to determine the weight of the crown. After all, if silver is mixed into it, then its weight would be different from what it would be if it were made of pure gold. The specific gravity of gold was known. But how to calculate the volume of the crown? After all, it had an irregular geometric shape.
According to legend, one day Archimedes, while taking a bath, was thinking about a problem that he had to solve. Suddenly, the scientist noticed that the water level in the bath became higher after he plunged into it. As it rose, the water level dropped. Archimedes noticed that with his body he was displacing a certain amount of water from the bath. And the volume of this water was equal to the volume of his own body. And then he understood how to solve the problem with the crown. It is enough just to immerse it in a vessel filled with water, and measure the volume of water displaced. They say that he was so delighted that with a cry of "Eureka!" (“Found it!”) jumped out of the bath without even getting dressed.
Whether this was true or not is irrelevant. Archimedes found a way to measure the volume of bodies with complex geometric shapes. He first drew attention to the properties physical bodies, which are called density, comparing them not with each other, but with the weight of water. But most importantly, he was open buoyancy principle .
Law of Archimedes
So, Archimedes established that a body immersed in a liquid displaces such a volume of liquid that is equal to the volume of the body itself. If only part of the body is immersed in the liquid, then it will displace the liquid, the volume of which will be equal to the volume of only the part that is immersed.
And a force acts on the body itself in the liquid, which pushes it to the surface. Its value is equal to the weight of the liquid displaced by it. This force is called the power of Archimedes .
For a liquid, Archimedes' law looks like this: A body immersed in a fluid is subjected to an upward buoyant force equal to the weight of the fluid displaced by the body.
The magnitude of the Archimedes force is calculated as follows:
F A = ρ ɡ V ,
where ρ is the density of the liquid,
ɡ - acceleration of gravity
V - the volume of a body immersed in a liquid, or the part of the volume of a body below the surface of the liquid.
The force of Archimedes is always applied to the center of gravity of the volume and is directed opposite to the force of gravity.
It should be said that for the fulfillment of this law one condition must be observed: the body either intersects with the boundary of the liquid, or is surrounded on all sides by this liquid. For a body that lies on the bottom and hermetically touches it, Archimedes' law does not apply. So, if we put a cube on the bottom, one of the faces of which will be in close contact with the bottom, we will not be able to apply the law of Archimedes to it.
The force of Archimedes is also called buoyancy force .
This force, by its nature, is the sum of all pressure forces acting from the side of the liquid on the surface of the body immersed in it. The buoyant force arises from the difference in hydrostatic pressure at different levels of the liquid.
Consider this force on the example of a body that has the shape of a cube or parallelogram.
P2- P 1 = ρ ɡ h
F A \u003d F 2 - F 1 \u003d ρɡhS \u003d ρɡhV
Archimedes' principle also applies to gases. But in this case, the buoyant force is called lifting, and to calculate it, the density of the liquid in the formula is replaced by the density of the gas.
Body floating condition
The ratio of the values of gravity and the force of Archimedes determines whether the body will float, sink or float.
If the Archimedes force and the force of gravity are equal in magnitude, then the body in the fluid is in a state of equilibrium when it does not float or sink. It is said to float in liquid. In this case F T = F A .
If the force of gravity is greater than the force of Archimedes, the body sinks or sinks.
Here F T ˃ F A .
And if the value of gravity is less than the force of Archimedes, the body floats. It happens when F T˂ F A .
But it does not emerge endlessly, but only until the moment when the force of gravity and the force of Archimedes are equal. After that, the body will float.
Why don't all bodies sink
If you put two bars of the same shape and size into water, one of which is made of plastic and the other of steel, you can see that the steel bar will sink, while the plastic one will remain afloat. It will be the same if you take any other objects of the same size and shape, but different in weight, for example, plastic and metal balls. The metal ball will sink and the plastic one will float.
But why do plastic and steel bars behave differently? After all, their volumes are the same.
Yes, the volumes are the same, but the bars themselves are made of different materials that have different densities. And if the density of the material is higher than the density of water, then the bar will sink, and if it is less, it will float until it is on the surface of the water. This is true not only for water, but also for any other liquid.
If we denote the density of the body P t , and the density of the medium in which it is located, as Ps , then if
P t Ps (the density of the body is higher than the density of the liquid) - the body sinks,
P t = Ps (the density of the body is equal to the density of the liquid) - the body floats in the liquid,
P t ˂ Ps (the density of the body is less than the density of the liquid) - the body floats until it reaches the surface. After which it floats.
The law of Archimedes is not fulfilled even in a state of weightlessness. In this case, there is no gravitational field, and, hence, the acceleration of free fall.
The property of a body immersed in a liquid to remain in equilibrium without rising or sinking further is called buoyancy .
