[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 71
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You can use as many signs as you like, as long as the equality is written correctly?
Oh, I guess this can only be checked on computer - if you emulate syntax validation. I think VB has such operator like eval( string expression )...
Знаки можно пользовать в любом количестве, лишь бы равенство было корректно написанным?
Ой, это, наверно, можно только на компутере проверить - если эмулировать проверку правильности синтаксиса.
Anything is possible. I haven't been able to disprove it.
On a computer, you have to program in all the possible ways of arranging the digits. There are as many as there are combinations of digits.
This has to be solved through number theory.
The theory of numbers (you mean integers?) is hardly helpful here. Only the computer. And logic, of course.
The hypothesis can be rephrased a little more simply: a series of 7 digits, if you use all the signs you mentioned, excluding equality, can always be converted to zero.
Летающая тарелка. Чуть выше облаков.
1. Стиральная машина.
2. Надо вызвать мастера.
No . It is supposed to be a perpetuum mobilee . Part 1 divides the hollow rim hermetically and moves freely at the top point. The air pressure on the right is greater than the pressure on the left, so the fluid level on the left is higher than the level on the right, so you get a moment of force proportional to the difference in levels.Во-первых, Вы противопоставляете статику и классическую механику. С чего бы это ?
Во-вторых, объясните откуда взялось требование: сумма всех сил, действующих на тело, равна нулю - то есть откуда взялось уравнение балланса. Почему эта сумма должна быть равна нулю ? Бывают ли случаи, когда она не равна нулю и что это за случаи ?
Получается, что в Вашем подходе присутствует два базовых утверждения: присутствие силы величиной -ma и требование нулевого баланса сил. А в механике Ньютона - только одно: уравнение второго закона.
Приведите пример, когда тело движется по направлению, нормальному траектории. Чему в этом случае равны центробежная и центростремительная силы ? Как вообще образуется траектория ? Почему иногда при равенстве равнодействующей нулю тело движется равномерно и прямолинейно, а иногда равномерно по окружности ?
Чтобы было понятно почему спрашиваю.
Тело никогда не движется по направлению, перпендикулярному траетории. Языком математики: вектор скорости тела всегда направлен по касательной. Если бы это было не так, то тело просто не двигалось бы по той линии, которую Вы называете траекторией. Оно двигалось бы по другой линии - в направлении скорости, т.е. по той линии, которую мы называем траеторией.
Хоть Вы и не ответили на мой вопрос о происхождении центробежной силы и следующие, я на Ваши отвечаю.
Авто наклоняется потому, что при повороте колес под углом к направлению скорости возникает пара сил, которая создает вращающий момент. Под действием этого момента авто наклоняется до тех пор, пока вращающий момент веса авто, направленный в противоположную сторону его не скомпенсирует.
Причина скольжения заключается в том, что сила трения колес об асфальт оказывается недостаточной, чтобы искривить траекторию авто в соответствии с кривизной шоссе. Авто движется по траетории той кривизны, которую способна создать возникающая сила трения. И это можно рассчитать и предсказать еще до вхождения в юз.
А как это в Вашем подходе ?
Успехов.
1. I am not contrasting classical mechanics with statics: what I am talking about is mechanics, including classical mechanics, as it does not contradict it. And forces of inertia exist and represent the property of bodies to resist a change in velocity or direction of initial motion. Otherwise, a car would stop immediately when you turn off the engine and press the brake and the passengers would not move forward, in the direction of the previous movement. A train, hitting a car at the crossing, provided the driver turned off the engine or disabled the torque transfer to the wheels, would stop immediately, instead of running over this car and the passengers would continue to sleep peacefully in their seats. There are plenty of examples. And also, as it is my speciality ;), I will tell more in detail: I remember as on a second year we were sounded fictitiousness of centrifugal force - simply then we had no bases to understand more, but it is the same that for simplicity of studying geography to begin to consider that the Earth is flat and is supported by 3 elephants - to solve the elementary problems will quite go ;) - In the subsequent 3.5 years (they study at our university for 6, i.e. 5.5 years) it has turned out not so and, unlike those who knows elective, I know that these "fictitious" forces are capable to make real work)
2. There are no cases where the balance equation is non-zero. There are cases where the sum of forces is non-zero, but this means that not all forces are taken into account.
3 The body, according to classical mechanics, will always tend to move not in the direction of the velocity vector but in the direction of the applied force, though not immediately, because it has inertia, which leads to the appearance of internal stresses (test question - what does the term VAT mean in mechanics?), but it is already beyond the Newtonian mechanics. The body moves uniformly and linearly in the inertial system, provided that no external forces are applied to the body.
4. About the car - the answer does not count. What is this pair of forces, where are they directed, where are they applied ? The next question will be about the nature of these forces, so you can go right ahead.
5. I wrote above about the origin of centrifugal force, I can repeat - it is the resistance of a body possessing mass to attempts to change the direction of the original trajectory of motion.
Good luck.
There are plenty of cases where a body moves normally on its original trajectory, mostly accidents, car skidding, e.g. turbine rotor collapse, the most obvious of the accident free ones is the slingshot.
ZZZY
Newton's third law of mechanics - In an inertial reference frame, the acceleration that a material point receives is directly proportional to the net force applied to it and inversely proportional to its mass - this is not one equation, it is an integral relationship. When you decompose the coordinate system into orthodes, and you get a system of equations. By the way, the 2nd law rewritten in the form ma+F=0 is the dynamic balance equation (for statics, there is no inertia, so just consider that the sum of all forces is zero, for resilience there is the elastic force: F=-kx - Hooke's law, for the dynamics of deformable bodies with energy dissipation, and the inertial force, for systems without dissipation - just the inertial force - and in general all the models are special cases of expansion of the sum of forces in Taylor series - when they integrated by hand they preferred Fourier). Therefore, in mechanics one prefers to use the equivalent treatment - Sum of all forces applied to a body is zero, with ma representing the force of inertia and hence the equation of motion or vibration is derived if the system is fixed so that it cannot move freely as a solid.
ZZZY Maybe somewhere not strictly formulated - I tried to explain "on the fingers".
Блин тоже не удержался . Вопрос 1 Для чего это нужно . Вопрос 2 Почему не работает .
Bah, that's where my equivalent fly lives :) . Then I'll add another question: what is it doing there? I mean at the highest point.
P.S. If it's a fly, maybe it's an aeroplane. :) But if it doesn't work, then it never took off. :(
2 Yuri and Vladislav:
While you are talking about the same mathematics (racing the same term from left to right or right to left) your argument looks pure scholasticism.
Vladislav, maybe you as the person who has experience of "non-inertial approach", simply give an example when the assumption about force of inertia (that is use of non-inertial system of readout) gives simpler mathematics? Surely it is clear to Jury that passengers simply try to continue uniform and rectilinear movement, but the carriage accelerates towards them under the force of friction.
After all, Copernicus won precisely because he proposed a simpler mathematics to describe the apparent motion of celestial bodies.
And the analogy with the force of elasticity is very lame, imho. Because if you remove the external force, the force of inertia disappears without a trace, but the force of elasticity can easily kick in the nose :)
Ба, да там же моя эквивалентная муха живёт :) . Тогда добавлю ещё вопрос: что она там делает? Я имею в виду в самой верхней точке.
Serves as an airtight partition so that the air pressure in the right-hand side is always greater than the pressure in the left-hand side, hence the different fluid levels and consequently the momentum of the forces.
2 Юрий и Владислав:
Пока вы говорите об одной и той же математике (гоняете один и тот же член слева-направо или справа-налево) ваш спор выглядит чистой воды схоластикой.
Владислав, может Вы как человек, имеющий опыт "неинерциального подхода", просто приведёте пример когда предположение о силе инерции (то есть использование неинерциальной системы отсчёта) даёт более простую математику? Наверняка ведь Юрию понятно, что пассажиры просто пытаются продолжить равномерное и прямолинейное движение, вот только вагон ускоряется им навстречу под действием силы трения.
Коперник ведь победил именно потому, что предложил для описания видимого движения небесных тел более простую математику.
А аналогия с силой упругости сильно хромает, имхо. Потому что если Вы уберёте внешнюю силу, сила инерции исчезнет бесследно, а вот сила упругости запросто может и в нос дать :)
It's not really an argument. It's just something to talk about.
The use of the force of inertia allows one to solve problems which are not solvable without its use - for example, the problem of collapse or heating of a body under mechanical loads. The Newtonian mechanics without taking into account the inertial terms will allow to determine these loads - all other things are not, obtaining the spectra of natural frequencies to determine the dangerous/non dangerous resonance zones, however here we intrude into the zone of theories of elasticity, plasticity, are faced with a mass of hypotheses, confirmed mainly empirically, but giving results..... mathematics thus does not become easier, especially in consideration of energy dissipation - there we generally leave the parabolic diffurs. But even in these theories, Newton's 2nd law is basic in one form or another.
Concerning the elasticity force: you're wrong - it appears under the influence of external forces - remove them (external forces) and we obtain that the elasticity force is zero - the body has no loads, i.e. there is nothing to solve: no external forces - the body is at rest or in uniform motion in the inertial system, which is equivalent. Add consideration of the force due to the body's mass and you will understand why a lift, for example, cannot be infinite-storeyed (cables will break under the impact of the structure's own weight - the strength limit is exceeded), etc.
Good luck.
SZZ the fact that Yury understands - I'm more than sure: just his phrase about absence of inertia force gives a possibility to abstract away more widely in some way from tasks of trading - after all that is the purpose of this thread ;).
to VladislavVG
I am a physicist by training, my specialty is theoretical physics. So you do not need to make an effort to explain it to me on my fingers.
And you, apparently, are a mechanic? If I'm wrong, correct me, it's interesting to know in what university and in what specialty they teach this way.
When I spoke about the opposition, I meant this phrase of yours:
Равенство нулю уравнения баланса не является условием статики - это классическая механика. ... В классическом виде записывается как равенство нулю суммы всех сил, действующих на тело.
Equality to zero of the sum of all forces acting on a body is a condition of equilibrium in statics. But you object to it.
Inertia, i.e. the property of bodies to resist, etc., is to me quite explainable and is a consequence of the law of conservation of energy-momentum. This law is quite clear even at the level of primitive intuition, just like the law of causality. It is the basis of all three laws of Newton. It is possible, of course, to consider that the force of inertia exists at least because it (as you write) is capable of producing work. Or you can simply be aware of the law of conservation of energy and see how kinetic energy is converted to potential energy, or thermal energy, or electromagnetic energy. And I guarantee you that there is no theoretical or practical problem that can be solved by "fictitious forces" and cannot be solved without them, i.e. classical mechanics.
By the way, by classical mechanics, I mean what was read to us - there were no "fictitious forces" anywhere, including Landau-Lifshitz "Course of theoretical physics". There was simply no need for them.
...
Therefore, in mechanics one prefers to use the equivalent treatment - Sum of all forces applied to the body is zero, with ma representing inertial forces and hence the equation of motion is derived...
I think the approach you advocate, has its roots in the cited quote. Statics, as we know, is simpler than dynamics. That's why mechanics came up with such a "simplification" - to present the equation of Newton's 2nd law in the form of the static force balance equation. Well, they had to introduce inertial force and say that it = -ma, what's wrong with that? It became easier to perceive the problems of mechanics, and thank God.
Thank you for enlightening me, but I would like to make some corrections. It's not an integral relationship, it's a differential relationship - only a differential relationship can determine the trajectory at each of its points, an integral relationship can't do that. It is indeed a system of equations, only it is wider than what you have described. If there are N bodies in the system, it is a system of N vector equations. When you decompose each of them along the coordinate axes you get a total of 3*N equations.
I think you already know this, and you just used the word "integral" in a different sense. So just consider that I have dotted the i's.
I don't really see the point of this discussion. I just do not understand your desire to present this approach of mechanics as something true, as real classical mechanics, unlike the one that we know, based on three elephants. some dissonance in it - it stings to the ear.
PS
About the auto - the answer doesn't count. What's with the power couple ...
Is it a value added tax ? :-)
I see that you decided to act as an examiner ? Too bad I forgot my report card at home.
I don't know what VAT is, so I'll probably lose a lot in your eyes now. But what can you do... But now I do not need to tell you what a pair of forces (this is a physical term - "pair of forces"), the momentum of force, where it comes from and where it is directed. Two is as good as two.
PPS
1. The equality of centrifugal and centripetal forces is zero as long as the body is not moving in a direction normal to the trajectory.
2. The cases in which a body moves normal to the original trajectory are mass, ...
Note these two formulations. Both belong to you. See the difference?
Apparently you're referring to a sudden change in trajectory? Or just a curvature of the trajectory with respect to the tangent (= velocity vector) at a given point? It is difficult to think that you meant the situation when the trajectory forms a right angle - it does not happen in life. It's not the wording - it's hard to understand.