Many properties of solids and liquids that we deal with in everyday life depend on their density. One of the accurate and at the same time simple methods for measuring the density of liquid and solid bodies is hydrostatic weighing. Consider what it is and what physical principle underlies its work.
Archimedes' Law
It is this physical law that forms the basis of hydrostatic weighing. Traditionally, its discovery is attributed to the Greek philosopher Archimedes, who was able to identify the fake gold crown without destroying it or doing any chemical analysis.
It is possible to formulate the law of Archimedes as follows: a body immersed in a liquid displaces it, and the weight of the displaced liquid is equal to the buoyancy force acting on the body vertically.
Many have noticed that it is much easier to hold any heavy object in water than in air. This fact is a demonstration of the action of the buoyancy force, which is alsocalled Archimedean. That is, in liquids, the apparent weight of bodies is less than their real weight in air.
Hydrostatic pressure and Archimedean force
The cause of the buoyancy force acting on absolutely any solid body placed in a liquid is hydrostatic pressure. It is calculated by the formula:
P=ρl gh
Where h and ρl are the depth and density of the liquid, respectively.
When a body is immersed in a liquid, the marked pressure acts on it from all sides. The total pressure on the side surface turns out to be zero, but the pressures applied to the lower and upper surfaces will differ, since these surfaces are at different depths. This difference results in a buoyancy force.
According to the law of Archimedes, a body immersed in a liquid displaces the weight of the latter, which is equal to the buoyant force. Then you can write the formula for this force:
FA=ρl Vl g
The symbol Vl denotes the volume of fluid displaced by the body. Obviously, it will be equal to the volume of the body if the latter is completely immersed in the liquid.
The Archimedes force FAdepends on only two quantities (ρl and Vl). It does not depend on the shape of the body or on its density.
What is a hydrostatic balance?
Galileo invented them at the end of the 16th century. A schematic representation of the balance is shown in the figure below.
In fact, these are ordinary scales, the principle of operation of which is based on the balance of two levers of the same length. At the ends of each lever there is a cup where loads of known mass can be placed. A hook is attached to the bottom of one of the cups. It is used for hanging loads. The scale also comes with a glass beaker or cylinder.
In the figure, the letters A and B mark two metal cylinders of equal volume. One of them (A) is hollow, the other (B) is solid. These cylinders are used to demonstrate Archimedes' principle.
The balance described is used to determine the density of unknown solids and liquids.
Hydrostatic weighing method
The principle of operation of scales is extremely simple. Let's describe it.
Suppose we need to determine the density of some unknown solid having an arbitrary shape. To do this, the body is suspended from the hook of the left scale and its mass is measured. Then water is poured into the glass and, placing the glass under a suspended load, it is immersed in water. The Archimedean force begins to act on the body, directed upwards. It leads to a violation of the previously established balance of weights. To restore this balance, it is necessary to remove a certain number of weights from the second bowl.
Knowing the mass of the measured body in air and water, as well as knowing the density of the latter, you can calculate the density of the body.
Hydrostatic weighing also allows you to determine the density of an unknown liquid. For thisit is necessary to weigh an arbitrary weight attached to a hook in an unknown liquid, and then in a liquid whose density is precisely determined. The measured data is sufficient to determine the density of the unknown liquid. Let's write the corresponding formula:
ρl2=ρl1 m2 / m 1
Here ρl1 is the density of a known liquid, m1 is the measured body mass in it, m2 - body mass in an unknown liquid, the density of which (ρl2) needs to be determined.
Determination of the fake gold crown
Let's solve the problem that Archimedes solved more than two thousand years ago. Let's use hydrostatic weighing of gold to determine if the royal crown is fake.
Using a hydrostatic balance, it was found that the crown in air has a mass of 1.3 kg, and in distilled water its mass was 1.17 kg. Is the crown golden?
The difference in the weights of the crown in air and in water is equal to the buoyancy force of Archimedes. Let's write this equality:
FA=m1 g - m2 g
Let's substitute the formula for FA into the equation and express the volume of the body. Get:
m1 g - m2 g=ρl V l g=>
Vs=Vl=(m1- m 2) / ρl
The volume of the displaced liquid Vl is equal to the volume of the body Vs as it is completely immersed inwater.
Knowing the volume of the crown, you can easily calculate its density ρs using the following formula:
ρs=m1 / Vs=m 1 ρl / (m1- m2)
Substitute the known data into this equation, we get:
ρs=1.31000 / (1.3 - 1.17)=10,000 kg/m3
We got the density of the metal the crown is made of. Referring to the density table, we see that this value for gold is 19320 kg/m3.
Thus, the crown in the experiment is not made of pure gold.
