One of the laws of hydrostatics is the rule of Archimedes. In the article we will tell you what it is, we will derive its formula. Consider what forces act on a body when it is completely and partially immersed in a liquid. Let's tell the story that helped Archimedes make his famous discovery.
Immersion of the body in liquid
Before moving on to the law of hydrostatics, let's do an experiment. We will weigh the body, for example, a bar or a piece of plasticine, using a dynamometer.
Weight is the force of the body on the suspension or support. In our case, the suspension is a dynamometer hook. The division price of the device is 0.05 Newton (N). Let's hang the body to it and see on the scale how much it weighs. The device shows a value of 1 N.
If the dynamometer shows a force of one newton, then it is pulled up by a force equal to one newton (spring force). Let's denote it with the letter F. The body is in balance, but what balances F? Gravity. It is attached to the center of gravityand directed downward. Fstrand=F=1 N.
Take a glass of water and gradually immerse the body in it (see the picture above). What happens to the dynamometer? As soon as the body has just touched the surface of the water, the dynamometer already shows a lower value (before immersion - figure a, after - figure b). The deeper the body sinks, the lower the dynamometer readings become. When the whole body is in the water, we will see a value of 0.2 Newton on the scale of the device.
Scheme will help us to understand the law of hydrostatics. Let's depict a dynamometer and a body in a liquid.
The spring of the device is stretched, as we found out, by a force of 0.2 N. Let's denote it F'. It is still pointing up because the dynamometer spring is under tension. When we immersed a body in a liquid, did the force of gravity acting on it change? No, the earth still attracts this body. Let's show on the diagram this force with the same vector as before.
Why then did the dynamometer readings decrease? In addition to gravitation and elasticity of the spring, now the upward buoyancy force Fvyt acts on the body from the side of the water. It is also called Archimedean (FА).
Let's find out what it is equal to in our case. To do this, we write down the equilibrium condition: upward F' and Fvyt together are balanced by gravity Fheavy. F' + FA =Fheavy. FA =Fheavy - F'. Let's establish by this formula what the force of Archimedes is equal to.FA=1 - 0, 2=0.8 N. We conducted the experiment, and now we will explain why this happens, what is the nature of this force.
Let's imagine a liquid in which the body is completely immersed. In depth, it is compressed, there is a pressure that is called hydrostatic. Its value depends on the depth and density of the liquid. The body in space occupies some volume. Its upper part is at a shallower depth, which means that the hydrostatic pressure there will be less than at the bottom. The lower body is under the most pressure.
To find the impact force, you need to multiply the pressure by the surface area of the body. If the pressure from above is less, then the force will be small. Let's denote it F1. The force acting on the bottom surface is F2. F2 > F1 because h2 (depth at the bottom of the bar) > h 1 (depth of upper body).
Pressure forces also act on the sides of the object. But since they are the same and directed in different directions, they compensate each other. The resultant F1 and F2 can be found by subtracting the smaller force from the larger one. F=F2 - F1. F is directed upwards, because the resultant of opposite forces always has the same vector as the largest of them. It will be impossible to derive a formula for the law of hydrostatics without this understanding.
The resultant F is the Archimedean force. FA =F2 - F1. Why is there a buoyant force? With increasing depth, the fluid pressureincreases. If we take atmospheric, it also depends on altitude. Every 12 m it decreases by 1 mmHg. That is why the balloon is always going up.
Calculation of buoyancy force
Not only the rule of Archimedes is one of the basic laws of hydrostatics. Pascal's law is also one of them. We will use it to derive a formula for finding the Archimedean force in case the body is not completely immersed in the liquid, but partially. Suppose we have the same body in the form of a rectangular parallelepiped, and it is immersed in a liquid. The area of the base of the body will be denoted by the letter S, and the depth to which the body was immersed, by the letter h. Let's draw a diagram that will help us make the calculation.
What forces act on the body? Above is atmospheric pressure. Let's denote the impact as P1. P1 =Patm. Let's call the pressure at depth P2. What does it equal? Atmospheric pressure also acts on the surface of the liquid. If it weren't there, then P2 would be just a hydrostatic action, which is calculated by the formula P=ρgh. But there is also atmospheric pressure. Pascal's law in hydrostatics states that the action on the fluid is transmitted to all its points, and this occurs without change. Atmospheric pressure is added to hydrostatic pressure. So P2 =Patm + ρg h.
Now we can find the force of pressure. From above, F1 presses on the body, from below - F2. The resultant of these forces will beArchimedean. F1 =P1 S or F1 =Patm S. F2 =P2 S or F2 =(P atm + ρgh)S. FA =F2 - F1. Substitute the data. FA =Patm S + ρghS - PatmS We reduce Patm S. This means that it does not matter what the atmospheric pressure is, the buoyancy force does not depend on it. But which indicator is important? This expression hS is the volume of the immersed part of the body. Let's denote it Vdip.
Experience of Archimedes
The basic law of hydrostatics is the rule of Archimedes: if a body is immersed in a liquid, it will displace a volume equal to the part of the body that is under the surface.
In Greece, King Hieron ruled. He ordered a gold crown from a jeweler to donate to the temple. He gave the master an ingot of gold, from which he made a crown. After a while, rumors reached Hieron that the jeweler had deceived him, replacing part of the metal with silver. The king invited Archimedes and asked him to check if this was true.
Archimedes went to the bath to freshen up. I must say that in ancient Greece, the bath was not a steam room, but a bathtub filled to the brim with cool water. Entering it, the scientist noticed that part of the liquid had spilled out. Moreover, the deeper Archimedes sank, the more water appeared on the floor. Thus, the discovery was made that the amount of liquid displaced is equal to the volume of the immersed body. Vdip =Vcutout.
Archimedes conducted the following experiment. Hetook a bar of gold of the same weight as the crown, and a bar of silver of the same weight. Archimedes immersed these ingots in a liquid. It turned out that silver displaces more water than gold. And when he immersed a crown of the same mass, it turned out that it displaces more liquid than a gold ingot, but less than a silver one. From this, Archimedes concluded that the jeweler was dishonest and added silver to the crown. He told Hieron about this, and he gave Archimedes a crown as a reward. What happened to the jeweler, history is silent.
Let's get back to the formula. Having made some transformations, we get FА =ρgVpogr. What is ρVdip? This is the mass of the displaced fluid. If we multiply it by the gravitational acceleration (ρgVpl), then we find out the force of gravity acting on the displaced fluid. But since the latter is motionless, this will be its weight.
Now we know that force is a vector that has a direction. It is directed upwards. The modulus of the vector is equal to the weight of the fluid displaced by the body. Based on this, it is possible to formulate the law of hydrostatics of Archimedes: a force equal to the weight of the liquid displaced by this body acts on a body that is lowered into a liquid. This rule is also called the principle of displacement.