Many of the properties of solids and liquids that we deal with in everyday life depend on their density. One of the exact 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 underlies hydrostatic weighing. Traditionally, his discovery is attributed to the Greek philosopher Archimedes, who was able to determine the fake of the golden crown without destroying it and without conducting any chemical analysis.
The following Archimedes law can be formulated: a body immersed in a liquid displaces it, and the weight of the displaced liquid is equal to the buoyant force acting on the body vertically upwards.
Many have noticed that it is much easier to hold a heavy object in water than in air. This fact is a demonstration of the action of the buoyancy force, which is also called Archimedean. That is, in liquids, the apparent weight of bodies is less than their real weight in air.
Hydrostatic Pressure and Archimedean Force
The reason for the appearance of a buoyant force acting on absolutely any solid placed in a liquid is hydrostatic pressure. It is calculated by the formula:
P = ρ l * g * h
Where h and ρ l are the depth and density of the fluid, respectively.
When the body is immersed in a liquid, then the marked pressure acts on it from all sides. The total pressure on the side surface is equal to zero, but the pressure applied to the lower and upper surfaces will differ, since these surfaces are at different depths. This difference leads to buoyancy.
According to the law of Archimedes, an immersed body in a liquid displaces the weight of the latter, which is equal to the buoyancy force. Then we can write the formula for this force:
F A = ρ l * V l * g
The symbol V l 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 strength of Archimedes F A depends on only two quantities (ρ l and V l ). It does not depend on the shape of the body or on its density.
What are hydrostatic scales?
At the end of the 16th century, Galileo invented them. A schematic representation of the balance is shown in the figure below.
In fact, these are ordinary scales, the principle 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. A glass cup or cylinder is also included with the scale.
In the figure, 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 the law of Archimedes.
The described scales are used to determine the density of unknown solids and liquids.
Hydrostatic weighing method
The principle of operation of the scales is extremely simple. We will describe it.
Suppose that we need to determine the density of some unknown solid that has an arbitrary shape. For this, the body is suspended from the hook of the left weighing pan and its mass is measured. Then, water is poured into the glass and, placing the glass under a suspended load, immerse it in water. An Archimedean force upward begins to act on the body. It leads to a violation of the previously established balance of the balance. 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 in water, as well as knowing the density of the latter, one can calculate the density of the body.
Hydrostatic weighing also allows you to determine the density of an unknown fluid. To do this, you need to weigh an arbitrary load, attached to a hook, in an unknown liquid, and then in a liquid whose density is precisely determined. The measured data are sufficient to determine the density of an unknown fluid. We write the corresponding formula:
ρ l2 = ρ l1 * m 2 / m 1
Here ρ l1 is the density of a known fluid, m 1 is the measured body mass in it, m 2 is the mass of a body in an unknown fluid whose density (ρ l2 ) must be determined.
Determination of fake gold crown
We will solve the problem that Archimedes solved more than two thousand years ago. We will use hydrostatic weighing of gold to determine the fake royal crown.
Using a hydrostatic balance, it was found that the corona 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 weight of the crown in air and in water is equal to the buoyancy force of Archimedes. We write this equality:
F A = m 1 * g - m 2 * g
We substitute the formula for F A and express the volume of the body. We get:
m 1 * g - m 2 * g = ρ l * V l * g =>
V s = V l = (m 1 - m 2 ) / ρ l
The volume of displaced fluid V l is equal to the volume of the body V s , since it is completely immersed in water.
Knowing the volume of the corona, one can easily calculate its density ρ s according to the following formula:
ρ s = m 1 / V s = m 1 * ρ l / (m 1 - m 2 )
We substitute the known data into this equality, we obtain:
ρ s = 1.3 * 1000 / (1.3 - 1.17) = 10 000 kg / m 3
We got the density of the metal the crown is made of. Turning to the density table, we see that this value for gold is 19320 kg / m 3 .
Thus, the crown in the experiment is not made of pure gold.