Archimedes' Law is a physical principle that states that a body that is immersed in whole or in part in a liquid is affected by a vertically directed force that is equal in magnitude 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 scholar Archimedes
Archimedes grew up in a family that was connected 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 friendship with Eratosthenes from Cyrene. Together with him, Archimedes first measured the circumference of the globe. Thanks to the influence of Eratosthenes, young Archimedes also showed 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 for a modern educated person.
Archimedes discovery of his law
According to historical information, Archimedes discovered his law in an interesting way. Vitruvius in his writings describes that the Syracuse tyrant Hieron II instructed one of the masters to cast him a golden crown. After the crown was ready, he decided to check to see if the master had tricked him and added cheaper silver, which has a lower density than the king of metals, to gold. He set this task to solve to Archimedes. The scientist could not violate the integrity of the crown.
Once taking a bath, Archimedes noticed that the water level in it rises. He decided to use this effect to calculate the volume of the corona, knowledge of which, as well as the mass of the corona, allowed him to calculate the density of the object. This discovery greatly struck Archimedes. Vitruvius described his condition like this: he ran down the street absolutely naked, and shouted “Eureka!”, Which is translated from ancient Greek as “I found!”. As a result, the density of the crown was 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 wording of the law of Archimedes, which was made by the scientist himself, has not practically changed.
The volume of fluid in equilibrium with the rest of the fluid
At school in the 7th grade, they begin to study the law of Archimedes. To understand the meaning of this law, you should first consider the forces that act on a certain volume of fluid that is in equilibrium in the thickness of the rest of the fluid.
The force acting on any surface of the considered volume of liquid is p * dS, where p is the pressure, which depends only on the depth, dS is the area of this surface.
Since the selected volume of liquid is in equilibrium, it means 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 resulting force is called the buoyancy force. Its application point is at the center of gravity of this volume of liquid.
Since the pressure in the liquid is calculated by the formula p = ro * g * h, where ro is the density of the liquid, g is the acceleration of gravity, h is the depth, the equilibrium of the volume of liquid under consideration is determined by the equation: body weight = ro * g * V, where V - the volume of the considered part of the liquid.
Solid Substitution
Considering further the Archimedes law in class 7 physics, we remove the liquid volume under consideration from its thickness, and put a solid body of the same volume and the same shape in free space.
In this case, the resulting pushing 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:
- The ejection force is ro * g * V.
- Body weight m * g.
In the simplest case, if the body is homogeneous, then its center of gravity coincides with the point of application of the buoyancy force.
The nature of Archimedes' law and an example of a solution for a body completely immersed in a liquid
Suppose that a homogeneous body of mass m was immersed in a fluid 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:
- The force is ro * g * x * S, 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 downward.
- The force is ro * g * (h + x) * S, which is associated with the pressure acting on the lower surface of the box. It is directed vertically upwards.
- Body weight m * g, which acts vertically down.
The pressure that the liquid creates on the lateral surfaces of the immersed body is equal in absolute value and opposite in direction, therefore, in total they give zero force.
In the case of equilibrium, we have: m * g + ro * g * x * S = ro * g * (h + x) * S, or m * g = ro * g * h * S.
Thus, the nature of the buoyancy force or the force of Archimedes lies in the difference in the pressures exerted by the liquid on the upper and lower surfaces of the body immersed in it.
Remarks on the law of Archimedes
The nature of the buoyancy force allows some conclusions to be drawn from this law. We give important conclusions and remarks:
- If the density of a solid is greater than the density of the liquid into which it is immersed, then the Archimedean force will not be enough to push this body out of the thickness of the liquid, and the body will sink. On the contrary, the body will float on the surface of the liquid only if its density is less than the density of this liquid.
- Under zero gravity conditions for volumes of liquid that cannot create a perceptible gravitational field on their own, there are no pressure gradients in the bulk of these volumes. In this case, the concept of buoyancy ceases to exist, and the law of Archimedes turns out to be 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.