Archimedes' Law is a physical principle that states that a body that is completely or partially immersed in a liquid is at rest acted upon by a vertically directed force, which in magnitude is equal to the weight of the liquid displaced by this body. This force is called hydrostatic or Archimedean. Like any force in physics, it is measured in newtons.
Greek scientist Archimedes
Archimedes grew up in a family that was associated with science, since his father, Phidias, was a great astronomer of his time. From early childhood, Archimedes began to show interest in the sciences. He studied in Alexandria, where he made friends with Eratosthenes of Cyrene. Together with him, Archimedes first measured the circumference of the globe. Through the influence of Eratosthenes, the young Archimedes also developed an interest in astronomy.
After returning to his hometown of Syracuse, the scientist devotes a large amount of time to the study of mathematics, physics, geometry, mechanics, optics and astronomy. In all these areas of science, Archimedes made various discoveries, the understanding of which is difficult even formodern educated person.
Archimedes discovers his law
According to historical information, Archimedes discovered his law in an interesting way. Vitruvius in his writings describes that the Syracusan tyrant Hieron II instructed one of the craftsmen to cast a golden crown for him. After the crown was ready, he decided to check if the master had deceived him, and if cheaper silver had been added to the gold, which has a lower density than the king of metals. He asked Archimedes to solve this problem. The scientist was not allowed to violate the integrity of the crown.
While taking a bath, Archimedes noticed that the water level in it was rising. He decided to use this effect to calculate the volume of the crown, the knowledge of which, as well as the mass of the crown, allowed him to calculate the density of the object. This discovery greatly impressed Archimedes. Vitruvius described his condition as follows: he ran down the street completely naked, and shouted "Eureka!", which is translated from ancient Greek as "I found it!". As a result, the density of the crown turned out to be less than pure gold, and the master was executed.
Archimedes created a work called "On Floating Bodies", where for the first time he describes in detail the law he discovered. Note that the formulation of the law of Archimedes, which the scientist himself made, has practically not changed.
The volume of liquid in equilibrium with the rest of the liquid
At school in the 7th grade, they begin to study the law of Archimedes. To understand the meaning of this law, we must first consider the forces that act ona certain volume of liquid that is in equilibrium in the thickness of the rest of the liquid.
The force acting on any surface of the considered volume of liquid is equal to pdS, where p is the pressure, which depends only on depth, dS is the area of this surface.
Since the selected volume of liquid is in equilibrium, it means that the resulting force acting on the surface of this volume, and associated with pressure, must be balanced by the weight of this volume of liquid. This resultant force is called the buoyancy force. Its point of application is in the center of gravity of this volume of liquid.
Since the pressure in a liquid is calculated by the formula p=rogh, where ro is the density of the liquid, g is the free fall acceleration, h is the depth, the equilibrium of the considered volume of liquid is determined by the equation: body weight=rog V, where V is the volume of the considered part of the liquid.
Replacing a liquid with a solid
Considering further the law of Archimedes in physics of the 7th grade, we will remove the considered volume of liquid from its thickness, and place a solid body of the same volume and the same shape in the free space.
In this case, the resulting buoyancy force, which depends only on the density of the liquid and its volume, will remain the same. The weight of the body, as well as its center of gravity, will generally change. As a result, two forces will initially act on the body:
- Pushing force rogV.
- Body weight mg.
In the simplest case, if the body is homogeneous, then its center of gravity coincides withpoint of application of the pushing force.
The nature of Archimedes' law and an example of a solution for a body completely immersed in a liquid
Assume that a homogeneous body of mass m is immersed in a liquid with density ro. In this case, the body has the shape of a parallelepiped with a base area S and a height h.
According to the law of Archimedes, the following forces will act on the body:
- Force rogxS, which is due to the pressure applied to the upper surface of the body, where x is the distance from the upper surface of the body to the surface of the liquid. This force is directed vertically downwards.
- Force rog(h+x)S, which is related to the pressure acting on the bottom surface of the parallelepiped. It is directed vertically upwards.
- The body weight mg that acts vertically down.
The pressure that the fluid creates on the side surfaces of the immersed body are equal in absolute value and opposite in direction, therefore they add up to zero force.
In case of equilibrium, we have: mg + rogxS=rog(h+x)S, or mg=roghS.
Thus, the nature of the buoyancy force or Archimedes force is the difference in pressure exerted by a liquid on the upper and lower surfaces of a body immersed in it.
Remarks on the law of Archimedes
The nature of the buoyancy force allows us to draw some conclusions from this law. Here are important conclusions and remarks:
- If the density of a solid is greater than the density of a liquid,into which it is immersed, then the Archimedean force will not be enough to push this body out of the liquid, and the body will sink. On the contrary, a body will float on the surface of a liquid only if its density is less than the density of this liquid.
- Under weightless conditions for liquid volumes that cannot create a perceptible gravitational field on their own, there are no pressure gradients in the thickness of these volumes. In this case, the concept of buoyancy ceases to exist, and the law of Archimedes is inapplicable.
- The sum of all hydrostatic forces acting on a body of arbitrary shape immersed in a liquid can be reduced to one force, which is directed vertically upward and applied to the center of gravity of the body. Thus, in reality there is no single force applied to the center of gravity, such a representation is only a mathematical simplification.
