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J. Perelman
"Entertaining physics". Book 2.
Chapter 6. Properties of liquids and gases

FANTASY AND MATHEMATICS

So tells the novelist; but no one will find if we check the facts referred to in this passage. We don't have to go down for this in the bowels of the Earth; for small excursions in the area of physics is enough to bring pencil and paper.

First of all we will try to determine at what depth you need to go down to the pressure of the atmosphere has increased by 1000 stake. The normal pressure of the atmosphere is equal to the weight of 760 mm column of mercury. If we were shipped into the air, and mercury, we would fall only on 760/1000 = 0.76 mm, so that the pressure is increased by 1000 stake. In the air, of course, we should fall for that much deeper, and just as many times, how many times the air easier mercury is 10 to 500 times. This means that the pressure is increased by 1000 stake normal, we must not fall to 0, 76 mm, as in mercury, and at 0, h, i.e. almost 8 PM When we will get another 8 m, the increased pressure will increase by another 1000 of its size, and so on*... At whatever level we were - at the ceiling of the world" (22 km), on top of mount Everest (9 km), or near the ocean surface, we you need to go down to 8 m, so that the pressure of the atmosphere has increased by 1000-th fraction of the original size. So, therefore, this table is the increase of pressure with depth:

On the Ground level pressure

  • 760 mm = normal
  • the depth of 8 m =1,001 normal
  • the depth 2x8 =(1,001)2
  • the depth 3x8 =(1,001)3
  • the depth of 4x8 =(1,001)4

And generally at a depth of p m the atmospheric pressure is greater than normal (1,001)n times; and until the pressure is very large, in the same time to increase the density of the air (the law of the MARRIOTT).

Note that in this case, as can be seen from the novel, about the hole in the Ground just 48 km, and therefore the weakening of gravity and the associated reduction in the weight of the air can not be taken into account.

Now you can calculate how big was about the pressure that underground travelers Jules Verne was tested at a depth of 48 km (48 000 m). In our formula, n is equal to 48000/8 = 6000. It is necessary to calculate 1,0016000. So as to multiply 1,001 self 6000 time - activity is quite boring and would take a lot of time, we will seek the help of logarithms. which rightly said Laplace that they are cutting the work, double the life of solvers**. Logarithmica are: the logarithm of the unknown is equal to

6000 * lg 1,001 = 6000 * 0,00043 = 2,6.

The logarithm of 2,6 find the desired number; it is 400.

So, at a depth of 48 km atmosphere pressure 400 times stronger than normal; the density of air under such pressure will increase, as shown by the experiments, 315 times. It is doubtful, therefore, that our underground travelers did not suffer, feeling only the pain in my ears"... In the novel by Jules Verne said, however, about reaching people more the depths of the underground, 120 and even 325 km air Pressure needed to reach there monstrous degrees; man is able to endure harmless for yourself an air pressure of not more than three or four atmospheres.

If by the same formula, we have to calculate at what depth the air becomes as dense as water, i.e. compacted in 770 times, you would dial: 53 km. But this result is incorrect, since at high pressures the density of the gas is not proportional to the pressure. The law of the MARRIOTT is quite accurate for a not too significant pressures not exceeding one hundred atmospheres. Here is data on air density, obtained on the experience:

Pressure, atmospheres Density
200 190
400 315
600 387
1500 513
1800 540
2100 564

The increase in density, as we can see, lags far behind the increase in pressure. In vain Jules-wernovsky scientist was expecting it reaches the depth where the air is denser than water, that he would not have to wait as the air reaches the density of water only under pressure of 3000 atmospheres, and then almost not compressed. About the same to turn the air in the solid state, one pressure, without the strongest cooling (below minus 146°), may not be speech.

Justice requires be noted, however, that the aforementioned novel by Jules Verne was published long before became known now given the facts. This justifies the author, though not correct narrative.

Let's use another described earlier formula to calculate the greatest depth of the mine, at the bottom of which people may remain without harm to their health. The highest air pressure, what is still able to move our body, - 3 ATM. Indicate the desired depth of the mine through x have the equation (1,001)x/8 = 3, where (logarithmica) computed X. we Get x = 8,9 km

So, people could without harm to be at a depth of about 9 km If the Pacific ocean suddenly dried up, people would be able to almost everywhere to live on its bottom.

* Following 8-foot layer of air is denser than the previous one, and therefore the increase of pressure is in absolute value larger than in the preceding layer. But it should be longer because it takes 1000-I share from larger values.
** Who got out of school lack of sense of logarithmic tables, that may change its hostile attitude towards them, having become acquainted with the characteristic, given the great French astronomer. There is this place in the Exposition of the system of the world": "the Invention of logarithms, reducing computation few months in the work of several days, as it doubles the life of astronomers and relieve them of their errors and fatigue, inseparable with long calculations. This invention so lestee for the human mind that is entirely borrowed from this source (i.e. the mind). In the technique of man to increase his power uses the materials and forces of nature; in the logarithms of all is the result of his own mind".

Entertaining physics J. Perelman

 




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