[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 61
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By convention, equality of velocities is maintained at all times, and this "forbids" forward motion. The trick is that in order to meet the condition of the problem, you have to apply some force (no matter what kind). For the simple reason that if you don't do this, the speed of the plane relative to the ground will in no way depend on the speed of the transporter. The wheels will simply spin faster and as a result, the condition will not hold.
As correctly noticed by Mathemat - the condition is incorrect (one can speculate about linear velocities - but the conditions don't) and within this incorrectness appear different solutions and as a consequence - different nicknames :o)
I have marked in black what is correct. Red - what's wrong. If you're not going to get indignant, you'll understand my explanation.
First of all, here's an example. Imagine you've squeezed a pencil between your palms. Now your task is to move your left hand down and your right hand up, but at exactly the same speed. All speeds are relative to your nose. Do you think you can move your left hand downwards. See if it can. How can it do that, if the movement of your right hand compensates 100% for the movement of your left hand?
Sergey, you're absolutely right that if you can't ignore the friction and mass of the wheels in this problem, then the answer is ambiguous. But even in this case, I can solve it and get the takeoff condition. It will contain as parameters the thrust force of the aircraft, the coefficient of friction in the landing gear, the mass and geometric dimensions of the wheels. I don't want to do it, of course - bummer, though. So let's try to handle a situation when there is no friction in chassis and moment of inertia of wheels is zero.
So, if the asphalt of the runway is standing, the point of contact of the wheel with the ground is stationary - there is friction with the ground. When the plane starts to accelerate, the wheels spin, but at every moment the point of contact with the ground is still stationary. Hence the relationship between the angular velocity of the wheels and the speed of the aircraft. Is that clear? Do you object?
Why does the plane accelerate? Because the thrust of the engines is acting on it. And what prevents the plane from accelerating? If we ignore the friction in the undercarriage system and the inertia of the wheels, then NOTHING !
Now imagine that the plane stands with the engines switched off, but the asphalt (transporter) is moving towards it. What do you think the plane will do ? You will be surprised - NOTHING ! For the same reason. There is no friction, there is no inertia in the wheels, so no force will be transmitted to the plane from the transporter that could move it. It will be stationary and the wheels will be spinning at the speed of the transporter.
Now all that's left to do is to combine the two processes. The result will be that the plane will take off as if nothing had happened, the transporter will move towards it with the same increasing speed relative to the ground and the wheels will be spinning with double acceleration.
If you understand and agree with it, you will agree that the force of friction in undercarriage and inertia of wheels of modern aircraft may prevent it from taking off under described conditions only if the plane is ill, has weak traction and does not want to fly.
а почему он не едет у Вас с включенным двигателем, при этом обутый в шасси, а не в ботинки? Обычно, если не включают тормоз (такая штука устанавливается на шасси, очень полезная), то самолет начинает потихоньку ехать.
who's who? ))))
Why doesn't it drive with the engine running, but with the landing gear on and not the boots on? Usually, if you don't turn on the brake (such a thing mounted on the landing gear, very useful), the plane starts to go slowly.
Farnsworth, why it doesn't go, I can't understand. It goes. If the engine develops the thrust necessary for this, it rides relative to the ground, and relative to the runway, it rides even faster. But the circumferential speed of its wheels is equal to the speed of the belt - that's the condition of the problem.
Почему все же самолет разгоняется ? Потому, что на него действует сила тяги двигателей. А что препятствует самолету ? Если мы пренебрегли трением в системе шасси и инертностью колес, то НИЧЕГО !
There is also the force of inertia of the aircraft itself (it has mass and accelerates - i.e. acceleration is different from zero) and the drag of the air (this is if we do not consider motion in a vacuum). But it has little effect - if the aircraft is moving (without considering conveyor belt motion) - it means that all drag forces are overcome, i.e. the resulting vector of forces is non-zero.
Good luck.
Выделил корень всего. Спор вокруг кпд возник из-за неправильного его определения. кпд = отношению полезной работы ко всей работе. Для кондиционера, который потребляя мощность 1квт дает комнате 1.5квт, полезная (т.е. переносимая извне) мощность составляет 0.5квт. Остальное - это потребляемая им мощность электроэнергии, которая в конечном счете преобразуется в тепло (как например у обычной нагревательной спирали). Поэтому для кондиционера кпд = 0.5/1.5 = 1/3. Больше единицы (в данном случае) кпд мог бы быть только если бы обеспечивал 2квт.
Т.о. ты абсолютно прав, коэффициент всегда меньше 2 и кпд всегда меньше 1.
No, it isn't. In refrigeration there is a parameter called efficiency. It has nothing to do with efficiency. Efficiency is the ratio of the amount of heat on the refrigerator circuit being heated to the energy input (not to be confused with power!). This parameter can be from 1 to... um... theoretically as much as you like. Its magnitude depends directly on the temperature difference between the cold and hot circuits, and tends to, let me put it neatly, to a very large number when the circuits are equalized in temperature. This is why steaming meat in the freezer causes a spike in heat release at the condenser. Inversely, the greater the temperature difference between the circuits, the efficiency drops, tending to 1. I.e. the heat release then tends to the amount of energy expended. In any case, efficiency and effectiveness are categorically different parameters. A refrigerator also has an efficiency, but is calculated slightly differently than a simple heat engine. It's all to be believed - I studied the subject once.
But, you wrote that it is a solution, it can give off heat energy by cooling down. That's where you're not quite right. When it cools, it does not give energy, it takes it from the outside environment. It would give it back if it were heating. The internal energy of the solution is greater, so it has no choice but to cool down to take energy from the external environment.
Richie, you have some kind of heat energy balance issue here.The body that takes in energy (no matter where it's coming from) gets hot. And the one that gives it away is cooling.
If it were like it's written in red, it would be a piece of cake to get warm in the frost - strip naked and enjoy it. :-)
...
Yuri, you'll be surprised, but I understand it all, I even simplified the problem significantly for you by writing a little earlier in the same post, the same conclusion - that the conveyor belt speed will not affect the take-off. The wheels will spin faster. I've been writing about this all along and I haven't argued at all. The direction of forces, their nature - believe me, I have a good idea :o). But all this time I was talking about something else entirely, about a not very correct condition, (not about a pencil). Maybe I misunderstood it somehow. If you will explain me exactly this condition, more precisely its possibility of occurrence - I will be grateful.
Any increase in conveyor belt speed - (quantitatively I could be wrong, sorry) doubles the rotation speed of the undercarriage, and this happens instantly (mind you - I'm not even arguing with that). It sort of just is. How do we enforce the condition of equality of web and chassis rotation speeds? (I remind you, what we've both come to - wheel rotation is always greater than belt speed).
PS: about its theoretical realization I do not speak yet.
Farnsworth, Почему не едит то, не могу понять. Едит. Если двигатель развивает тягу необходимую для этого, то едит относительно земли, а относительно полосы едит ещё быстрее. А вот окружная скорость колёс у него равна скорости движения ленты - это условие задачи.
It was a joke, actually.
to xeon
I'm not a vindictive person, but I do reread the logs.
:о)
There is also the force of inertia of the aircraft itself (it has mass and accelerates - i.e. acceleration is different from zero) and the drag of the air (this is if we do not consider motion in a vacuum). But it has little effect - if the aircraft is moving (without considering conveyor belt motion) - it means that all drag forces are overcome, i.e. the resulting vector of forces is non-zero.
Good luck.
"The force of inertia of the aircraft itself" does not exist in nature. And the drag of the air, while the aircraft is still in contact with the ground, can be neglected - too small due to the low speed of the aircraft.
Richie, you have some sort of heat energy balance issue here. The body that takes in energy (no matter where it's coming from) heats up. The one that gives it away is cooling.
If it were like it's written in red, it would be a piece of cake to get warm in the frost - strip naked and enjoy it. :-)
You're just mixing up the bodies. The internal energy of a solution is higher than that of its individual components, water and salt. Therefore, when the salt dissolves, it cools down. The mass of the substances is the same, but the energy of the solution is higher, so it turns out that in order for the solution to form, it has to absorb this energy from somewhere. And it can only absorb it by cooling down. The energy will come from the outside environment.