February 6th, 2012 , 04:51 pm

A seemingly hackneyed question: which is heavier?

Like a ton is just a ton. but not everything is so simple.

There are no instruments for measuring mass, there are only scales that measure weight. Let's say they put a ton of iron and a ton of cotton wool on large scales. We balanced it and marked “equal” in the notebook. Now we divided a pile of cotton wool and a bar of iron, say, into 1000 equal parts, respectively. We weighed them individually and what do we find? The total weight of cotton wool is noticeably greater. Hmm...how so?

We remember school physics, the laws of Archimedes. And we hardly remember, or look in the reference book, that the Archimedes force acts not only in liquids, but also in gases. It’s weak, of course, but it’s there. Air is a gas, a mixture of gases, and the law works in it. The larger the volume, the greater the Archimedes force, directed strictly against the force of gravity, which the scales measure. And because the volume of a heap of cotton wool weighing 1 ton is much greater than the volume of a ton of iron, hence the discrepancy. like this.

There is also a variation of this question about a kilogram of fluff and a kilogram of lead, and so on. But here is what Perelman writes:

There is a well-known joke question: what is heavier - a ton of wood or a ton of iron? Without thinking, they usually answer that a ton of iron is heavier, causing friendly laughter from those around them.

Jokers will probably laugh even louder if they are told that a ton of wood is heavier than a ton of iron. Such a statement seems incompatible with anything - and yet, strictly speaking, this answer is correct!

The fact is that Archimedes' law applies not only to liquids, but also to gases. Each body in the air “loses” as much of its weight as the volume of air displaced by the body weighs.

Wood and iron also, of course, lose some of their weight in the air. To get their true weights, you need to add the loss. Therefore, the true weight of the tree in our case is 1 ton + the weight of air in the volume of the tree; the true weight of iron is 1 ton + the weight of air in the volume of iron.

But a ton of wood occupies a much larger volume than a ton of iron (15 times), so the true weight of a ton of wood is greater than the true weight of a ton of iron! To put it more precisely, we would have to say: the true weight of that tree, which weighs a ton in the air, is greater than the true weight of that iron, which also weighs one ton in the air.

Since a ton of iron occupies a volume of 1/8 cubic. m, and a ton of wood is about 2 cubic meters. m, then the difference in the weight of the air displaced by them should be about 2.5 kg.

That's how much heavier a ton of wood actually is than a ton of iron! (Ya. Perelman “Entertaining Physics”. Book 1. Chapter 5. Properties of Liquids and Gases)

However, not everyone agrees with him. Do you agree? Read the opinions of opponents...

If you adhere to the false theory of classical physics, then Perelman’s conclusions are correct.
However, he did not know about the mistakes I mentioned above. That's why he created a paradox.
It is very easy to make sure that he is wrong.

Let's take two identical dynamometers and hang them on one piece of iron and on the other a piece of wood, as shown in Fig. 1

Fig.1
We will select the loads in such a way that both dynamometers show a weight value of 1 ton. Then we hang the dynamometers from the lever scales.
Since the weight of the dynamometers is the same, the arrow of the lever scales will be set to zero.
Thus, the correct answer to the question “Which is heavier, a ton of wood or a ton of iron?” is the following: The weight of a ton of iron is exactly equal to the weight of a ton of wood.

If buoyant force existed in reality, then the lever scale would show 2.5 kg. Luckily this doesn't happen!
The dynamometer takes into account all forces acting on the body. And if it shows 1 ton, then no other forces can act on this ton!!!

Read more about why there is no Archimedes buoyant force,

Well, another opinion from Internet readers:

It’s just that the mass of a body and its weight are not the same thing)) And if, by saying “heavier,” the mass of the object is meant, then wood and iron have the same mass, but different weights.

A ton is a unit of mass, measured in kilograms; weight is the force with which a body presses on a support, measured in newtons. Archimedean force is also measured in newtons and the above arguments apply to the weight of the body, because we are talking about the sum of two forces applied to the center of mass. The mass of one ton of wood is equal to the mass of one ton of iron. However, their weight will be different.

I don’t quite understand this theory, that is, if we start measuring the volume of this tree, then we also need to measure the volume of displaced air? What nonsense? air is a separate component, the air contained in the pores of the tree is already taken into account, the one that is displaced is a separate part, not connected in any way with the tree itself, but what if you measure the weight of an object in water? Do I also need to add displacement? that is, in fact, our ships weigh ten times more? I don’t understand, it seems like complete nonsense to me.

Is this really from Perelman? From childhood I remember that Perelman argued that a ton of iron is heavier, not a ton of wood.

“Heavier” is the weight with which the body being weighed presses on the scales, i.e. what the scales show. Archimedes' force DECREASES weight and wood in the atmosphere becomes LESS HEAVY, i.e. EASIER. Ton - unit. mass measurements, a ton of wood displaces a larger volume, and is LIGHTER than a ton of iron. A ton of fluff is even lighter, but a ton of helium balloons will generally show negative weight ;-)

I re-read it more carefully, the respected professor got a little weird - he weighs wood and iron in the air on a scale and names the weight in tons (error, weight in newtons), then offers to estimate the “true weight” by pumping out the air. I think that in the atmosphere and in water and in vacuum, the weight is always true; in determining the weight there is no condition to exclude external forces.

If you stand under the balcony, and I drop a kilogram of fluff on your head, and then a kilogram of iron, then you will feel heavier

According to the problem, we have a CLEARLY measured/weighed ton of iron and a ton of wood. Here volume no longer plays a role. But if, after measuring/weighing, we move these two compared objects vertically relative to sea level/weighing point, we will get a small discrepancy...

Some kind of bullshit. There is no true weight, there is mass, and there is weight. Weight is the force of pressure on a support. If you weigh to get the same weight, then the mass of the wood will be greater, and if you take the same mass, then the weight of the iron will be greater. Usually the tasks are simply not set correctly.

An absolute nightmare - due to the terrible confusion in terms. The word “mass” is missing altogether! After such “articles”, confusion arises in the head.

So, is Perelman wrong or not?

By the way, when asked why Perelman refused a million for proving Poincaré’s theorem, he replied:

“I know how to control the Universe. And tell me, why should I run for a million?”

Fragments of the interview

Grigory Yakovlevich, as a schoolboy you represented the USSR at the Mathematical Olympiad in Budapest. And they took the gold medal...

Preparing for the Olympiad, we tried to solve problems where the ability to think abstractly was an indispensable condition. This distraction from mathematical logic was the main point of daily training. To find the right solution, it was necessary to imagine a “piece of the world.”

Isn't it a little difficult for schoolchildren?

If we talk about conditioned and unconditioned reflexes, a baby experiences the world from birth. If you can train your arms and legs, then why can't you train your brain?

Do you remember any problem of that time that seemed unsolvable?

Unsolvable... Perhaps not. Difficult to solve. That's more accurate. Remember the biblical legend about how Jesus Christ walked on water as well as dry land. So I needed to calculate how fast he had to move through the waters so as not to fall through.

Were the calculations correct?

Well, if the legend still exists, then I was not mistaken. There is no particular mystery here. Thanks to our teachers, we have already studied topology quite well - a science that allows us to understand the properties of space and operate with formulas, understanding their applied significance, which helps to achieve quick and accurate results. By the way, at that time I did not consider winning the Olympiad to be some kind of significant event - it was just one of many stages of knowledge in my favorite science.

Could have become a musician

Do you know that I had to rack my brains when choosing a profession?

How so?

I had the right to enter any educational institution in the Soviet Union without exams. So I hesitated between the Faculty of Mechanics and Mathematics and the Conservatory. I chose mathematics... Now it’s very interesting for me to remember my student years. We managed to do so much then... The process of learning was exciting... We forgot about the days of the week and the time of year.

At the age of twenty, you said a new word in science...

I didn’t say any words... I just continued to explore the problems of studying the properties of the three-dimensional space of the Universe. It is very interesting.

Have you tried to embrace the immensity?

Absolutely right... But anything that is immense is also embraced. I wrote my dissertation under the guidance of Academician Alexandrov. The topic was not difficult: “Saddle-shaped surfaces in Euclidean geometry.” Can you imagine surfaces of equal size and unevenly spaced from each other at infinity? We need to measure the "valleys" between them.

Is this a theory?

This is already practice. In what orbit will the spaceship fly to the constellation Canis? What obstacles will you encounter along the way... Do you want it even easier? Is it worth cutting hay between three hills? How many people and cars are needed for this? The Ministry of Agriculture, it turns out, is of no use. There is a formula. Use it. Count. And you are not afraid of any crises.

Isn't this scholasticism?

This is a wheel, an axe, a hammer, an anvil - anything but scholasticism. Let's figure it out. The peculiarities of modern mathematics are that it studies artificially invented objects. There are no multidimensional spaces in nature, there are no groups, fields and rings, the properties of which are intensively studied by mathematicians. And if in technology new apparatuses and all kinds of devices are constantly being created, then in mathematics their analogues are being created - logical techniques for analysts in any field of science. And any mathematical theory, if it is rigorous, sooner or later finds application. For example, many generations of mathematicians and philosophers have tried to axiomatize philosophy. As a result of these attempts, the theory of Boolean functions was created, named after the Irish mathematician and philosopher George Boole. This theory became the core of cybernetics and the general theory of control, which, together with the achievements of other sciences, led to the creation of computers, modern sea, air and space ships. Such examples are the history of mathematics
gives tens.

Does this mean that each of your theoretical developments has practical significance?

Undoubtedly. Why did you have to struggle for so many years to prove the Poincaré conjecture? Simply, its essence can be stated as follows: if a three-dimensional surface is somewhat similar to a sphere, then it can be straightened into a sphere. Poincaré's statement is called the “Formula of the Universe” because of its importance in the study of complex physical processes in the theory of the universe and because it provides an answer to the question of the shape of the Universe. This evidence will play a big role in the development of nanotechnology.

This means “cheerful”, “life-affirming” reports from the “pioneers” of this industry...

Absolute nonsense and nonsense. An attempt to build a house on sand... I learned to calculate voids, together with my colleagues we are learning the mechanisms of filling social and economic “voids”. There are voids everywhere. They can be calculated, and this gives great opportunities... I know how to control the Universe. And tell me - why should I run for a million?!

BY THE WAY

Why else would they give a million dollars...

In 1998, with funds from billionaire Landon T. Clay, the Clay Mathematics Institute was founded in Cambridge (USA) to popularize mathematics. On May 24, 2000, the institute's experts selected the seven most, in their opinion, puzzling problems. And they assigned a million dollars for each.

1. Cook's problem

It is necessary to determine whether checking the correctness of a solution to a problem can take longer than obtaining the solution itself. This logical problem is important for specialists in cryptography - data encryption.

2. Riemann hypothesis

There are so-called prime numbers, such as 2, 3, 5, 7, etc., which are only divisible by themselves. It is not known how many there are in total. Riemann believed that this could be determined and the pattern of their distribution could be found. Whoever finds it will also provide cryptography services.

3. Birch and Swinnerton-Dyer conjecture

The problem involves solving equations with three unknowns raised to powers. You need to figure out how to solve them, regardless of complexity.

4. Hodge conjecture

In the twentieth century, mathematicians discovered a method for studying the shape of complex objects. The idea is to use simple “bricks” instead of the object itself, which are glued together and form its likeness. It is necessary to prove that this is always permissible.

5. Navier - Stokes equations

It’s worth remembering them on the plane. The equations describe the air currents that keep it in the air. Now equations are solved approximately, using approximate formulas. We need to find the exact ones and prove that in three-dimensional space there is a solution to the equations that is always true.

6. Yang - Mills equations

In the world of physics there is a hypothesis: if an elementary particle has mass, then there is a lower limit to it. But which one is not clear. We need to get to him. This is perhaps the most difficult task. To solve it, it is necessary to create a “theory of everything” - equations that unite all the forces and interactions in nature. Anyone who can do it will probably receive a Nobel Prize.

sources

Over time, excess metal forms in any enterprise, household or just a private home.

Household cast iron products such as batteries and baths, belong to the category 19A and are characterized by a high phosphorus content.

Separately, it is worth mentioning companies that are engaged in self-pickup of old batteries and - often the reward they offer underestimated several times. Or there is no payment at all, that is, the benefit of the owner of the scrap is the ridding of the home from metal that has served its useful life.

Rail scrap

Price for rail scrap brand 3AB on average is 8.5-12 rubles for 1 kg depending on the collection point. The price is affected by the percentage of technical blockage in the iron - the lower it is, the higher the cost.

The price of 8.5-12 rubles per 1 kg is indicated for scrap metal with 1.5% blockage, this is precisely the content of foreign impurities in most rail scrap.

Bearing scrap

Price for 1 kn of bearing scrap depends on the composition of the metal, from which they are made. Bearings are made from several metals:

• bronze,
• brass,
• steel
• babbitt.

If the bearing steel, then most likely passes under the brand name 3B3 and is made of high quality alloy steel. The price for such scrap metal is negotiated individually at the collection point or is set at the category level A3.

Income from the sale of niresist

Niresist- This is cast iron alloyed with nickel.

Due to the significant nickel content, the price of non-resist scrap metal is quite high - 30-45 rubles for 1 kg.

When delivering a ton or more, the prices for receiving a shipment increase by several rubles per 1 kg.

The larger the batch of niresist scrap, the higher the price offered to you.

Melting slag and shavings

Melting slag and shavings are assessed individually depending on the volume of the batch for delivery. The price remains in the range 7-10 rubles per kilogram.

Many collection points set restrictions and do not accept shipments weighing more than 100 kilograms.

Conclusion

As you can see, the price of ferrous scrap metal fluctuates from 7 to 12 rubles for 1 kilogram. The exception is such metals as chromium steel and ni-resist; the price of scrap for such ferrous metals reaches up to 45 rubles for 1 kg.

When delivering a ton or more, the cost increases by 1-2 rubles, and the larger the batch of metal, the more profitable it can be delivered. Also, when delivering ferrous metal, the price for 1 kg of metal depends not only on the brand and quality of the iron, but also on the specific collection point.

The lowest cost is provided by those points that are located in the regions of the country, while in Moscow and other large cities of federal significance they are higher.

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