In our life, from time to time we are confronted with tasks of ingenuity, the answer to which at first glance seems obvious, and sometimes the riddles themselves look silly. One of these is the following: "What is heavier - a kilogram of cotton wool or a kilogram of iron?". We give an answer to it in the article.
"Evidence" of the answer
After asking a question about which is heavier - a kilogram of cotton wool or a kilogram of iron, almost every person who is not familiar with physics well enough will answer that the substances in the problem have equal weight or equal mass, since 1 kilogram of some matter cannot be more or less 1 kilogram of other matter, because we are talking about one kilogram in both cases.
Nevertheless, if one thinks more thoroughly and recalls the law of Archimedes, then the issue under consideration will no longer seem so meaningless.
What does the law of Archimedes say?
This law is already being studied in the 7th grade of secondary schools. He claims that absolutely any solid body that is immersed in a fluid substance begins to act as a buoyant force. This force is called Archimedean, named after the ancient Greek philosopher Archimedes, who first discovered and described this effect.
In the wording of the law there are the words "fluid substance". This is the name of any bodies that are able to change their shape when exposed to infinitely small external forces. All liquids and gases satisfy this definition.
Thus, if a solid is placed in a liquid or gas, it will be pushed out of the corresponding substance. Archimedean force is determined by the following formula:
F A = ρ * V * g,
here ρ is the density of the substance, V is the volume occupied by the body immersed in it, g is the acceleration of gravity.
The reason for the occurrence of this force is the difference in pressure that acts on the upper and lower surfaces of a solid body immersed in a fluid substance, provided that the latter is in a nonzero gravitational field.
How is the law of Archimedes related to the issue under consideration?
To understand what is harder - a kilogram of wool or a kilogram of iron, you need to remember that the body weight is determined using weights (the type of weights does not matter), and all the bodies around us are in the air, which is the same fluid substance that appears in the law Archimedes. This means that any body that a person encounters in everyday life is affected by a buoyant force directed opposite to the body weight vector.
Therefore, when measuring with weights, we get not the true value of the body weight, but the difference between it and the strength of Archimedes. Then this difference is converted to mass, according to a simple formula:
P - F A = m * g,
where P is the weight, and m is the mass of the measured body.
Thus, it is possible to correctly determine what is harder - a kilogram of cotton wool or iron, if F A is calculated for these substances.
Correct answer
Considering the question of which is heavier - a kilogram of cotton wool or iron, the answer can be obtained using the above formula (P - F A = m * g) for cotton wool and for iron. Determine the true mass of each substance.
For cotton we have: P - F A = m 0 * g or m * g - ρ 0 * V * g = m 0 * g, reducing g, we get: m - ρ 0 * V = m 0 , where m 0 is the measured weights are the mass of cotton wool, that is, 1 kg, m is the true mass of cotton wool taking into account the buoyancy force, ρ 0 is the density of air, V is the volume of cotton wool.
Knowing that V = m / ρ, where ρ is the density of the cotton wool, and substituting this formula in the expression under consideration, we obtain: m - ρ 0 * m / ρ = m 0 or m = m 0 / (1-ρ 0 / ρ) . As can be seen from the expression, the real mass of cotton wool can be found if you know its density and air density. Turning to the reference data, we substitute ρ 0 = 1.225 kg / m 3 and ρ = 35 kg / m 3 (different types of watts have different densities, the highest value is taken for example), we obtain: m = m 0 / (1-ρ 0 / ρ) = 1 / (1-1,225 / 35) = 1,036 kg. That is, if the mass of cotton wool was determined in an airless space, then exactly the value of 1.036 kg would be obtained.
Similarly, we calculate the true mass of iron (ρ = 7874 kg / m 3 ): m = m 0 / (1-ρ 0 / ρ) = 1 / (1-1,225 / 7874) = 1,00016 kg.
As can be seen from the result, the mass of cotton wool is greater than that for iron under these conditions.
Why is a kilogram of cotton wool heavier than a kilogram of iron? Because the volume of 1 kg of cotton wool is greater than this value for 1 kg of iron, which means that the Archimedean force for cotton wool is large and has a stronger effect on its true mass.