Archimedes' principle

Archimedes' principle was formulated by the Greek scholar Archimedes over 2000 years ago. It reads:

The static buoyancy of a body in a medium is equal to the weight of the displaced fluid from the body.

Archimedes' principle applies to all fluids, i.e. liquids and gases. Ships displace water and will get a boost. Since the average density of the vessel is less than the density of water, it floats on the surface. Also balloons and airships make use of this property in order to go to. For this they are filled with a gas whose density is lower than that of the surrounding air. These gases (for example helium or hydrogen) are in airships and balloons many inherently less dense than air, hot air balloons, the air charge by means of gas burners is heated, whereby its density decreases.

Explanation of the phenomenon

Cause the lift force is caused by the gravitational pressure difference between the top and the bottom of a submerged body. The forces acting on the lateral surfaces, do not play a role, since they cancel each other out. That is, it acts on the lower parts of the surface of a submerged body has a greater strength than at the upper portions of the surface. Since each physical system is always striving to achieve a pressure balance, the body will move as long upward until balance all forces acting on it.

Example calculation

In the example ( Fig. 1) we assume a cube with edge length of 20 cm. He is deeply immersed 10 cm below the water surface.

On the lower surface (Fig. 1) the force acts

Upwards. To the top surface, however, the force acts

Down. The difference between the two forces, ie the buoyancy of this body, is 78.48 N.

According to Archimedes, the following applies: . Referring to the example ( Fig. 1) we can write:

The density of the fluid, the relationship to the mass and volume, and the local factor was used. We see that both methods give the same result.

Thought experiment

The following thought experiment illustrates the validity of Archimedes' principle. To this end, imagine a stationary fluid. Within the fluid, any portion of the fluid is marked. The label may be something like a water balloon imagine in a container of water, except that the skin of this water balloon is infinitely thin and massless and can take any form.

It is found now that the so- marked part of the fluid within the fluid neither increases nor decreases, since all of the fluid is at rest - the marked part seems to literally float weightlessly in the surrounding fluid. This means that the buoyancy of the selected portion of fluid accurately compensate its weight. It can be concluded that the buoyancy of the marked fluid part exactly equal to its weight. Since the mark is arbitrary within the fluid, thus the accuracy of the Archimedean principle for homogeneous fluids is shown.

Rising, falling, hovering

Thus, the body maintains the position described in the graph, its weight force of the weight of the displaced water ( 78.48 N) must be equal. Then all the forces acting on the body lift on and this comes to a standstill. According to the formula, the body must be 8,000 g. In addition, he was, after a density of 1 kg/dm.sup.3, so the density of water.

So we can formulate the following rule:

  • If it is, then the body floats.
  • If it is, then the body is increased.
  • If it is, then the body is reduced.

The bodies rise or fall until the weight force counteracts a magnitude equal force. This may result in a changing density of the fluid or the bottom of the cup during sinking. A body often rises until it breaks through the surface. In this case, the following applies:.

The discovery of Archimedes' principle

Archimedes was commissioned by king Hiero II of Syracuse to find out whether as ordered pure gold whose crown would be, or whether the material has been stretched by cheaper metal. This task was Archimedes first problems since the crown, of course, was not to be destroyed.

According to tradition, Archimedes finally had the saving idea when he rose to bathe in a tub filled to the brim and it ran through the water. He realized that the amount of water that had overflowed, exactly corresponded to his body volume. Allegedly, he then ran, naked as he was, through the streets and shouted Eureka ("I have found it ").

To solve the task, he plunged once the crown and then a bar of gold, which weighed as much as the crown, in a container filled to the brim water tank and measured the amount of the overflowing water. Since the crown displaced more water than the gold bullion and thus was more voluminous at the same weight, she did not have a material of lower density, ie of pure gold, may have been manufactured.

This story was narrated by the Roman architect Vitruvius.

Although the legend on this story due the discovery of Archimedes' principle according to which test of Archimedes would also work with any other liquid. The interesting thing about the Archimedean principle, namely the emergence of buoyancy and thus the calculation of the density of the fluid, in this discovery, history does not matter.

Physical derivation

A body is loaded by the pressure that the surrounding medium ( liquid or gas) has on its surface. An observed portion of the surface with the contents of which are so small that it is practically planar and that in its region of the pressure P is constant. Is the unit vector of the outer surface normal of the patch The medium then exerts the force

From the section. A summation of these forces about any cuts will supply all the buoyancy force.

Archimedes' principle applies only if and strict when the displaced medium is incompressible ( non-compressible ) is. Liquid such as water is well satisfied, therefore, to be understood in the following of a body immersed in a liquid the density (strictly dependent on the temperature).

In the liquid rests on a horizontal surface the size in depth the weight of a column of liquid bulk. The pressure at that depth is therefore

A corresponding pressure curve is valid at not too large differences in height z in the air or other gases (ie, the compressibility is of no consequence; case of large differences in height would have a variable density are taken into account ). Therefore, the following considerations apply to realistic large airships or balloons.

For simple geometric shapes, you can recalculate the validity of Archimedes' principle in simple terms by hand. For a cuboid with base area A and height h, which is immersed vertically into the liquid, obtained, for example:

  • Force to the upper base with the surface normal of:
  • Force on the lower base surface with the surface normal of:
  • Forces on the faces stand out always on each other.
  • The total buoyancy force is therefore

Here V is the volume displaced, so the displaced mass and its weight. Archimedes' principle is fulfilled. The negative sign is omitted if the z axis is chosen upward.

For an arbitrarily shaped body the entire buoyancy force is obtained by the surface integral

With the integral theorem

And it follows

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