On the image below you can see the electro motor with its own red axis, the green cable and the violet sphere attached to the cable. The cable is made of tensile material and it is stretched (never mind how and why). I need to rotate this cable+sphere but so that the cable to remain always in stretched condition, as if it is made of some rigid material (steel for example). Please Register or Log in to view the hidden image! In other words, the cable must not wrap around the electro motor’s own axis as shown on the image: Please Register or Log in to view the hidden image! How can I achieve this? Should I rotate the cable too slowly? Or should I attach additional violet rod to electro motor’s axis and then to attach the cable to that rod? I think that this additional violet rod would be useful and it will act like human’s hand when human rotates with opened hand and with cable/rope holding in his fist…… This structure (motor, cable, sphere) may be in space or at some celestial object-never mind. I simply would like to know how to rotate cable in stretched condition without any bend…..Please Register or Log in to view the hidden image!
As I understand it you will always have some bending, simply because A) there's no such thing as a perfectly stiff material and B) the rotation is driven from the rooted end of the cable, therefore there will be some lag before the free end rotates. If you use that violet rod all you are effectively doing is switching from a relatively inflexible material to a more flexible one, which will give an inflection point at the join. How much "bend" is unacceptable? Bearing in mind that if you rotate the thing fast enough any lag will be near enough invisible to the naked eye anyway? I think that the slower you rotate it the more noticeable will be the "bending".
Presuming that that angular velocity is constant, I believe the only real force that would bend the cable would be air resistance. Run this in a vacuum and I think you'll only see a bending during the initial angular acceleration phase. This is ignoring the "downward bend" caused by gravity, of course, which could be almost completely overcome by sufficient angular velocity. Someone may correct me on my assessment though so don't go making NASA components without a second opinion. Please Register or Log in to view the hidden image!
Dywyddyr Of course, in my case it will be elastic tether Let it be Please Register or Log in to view the hidden image! some bent will be tolerable, the point is that the cable MUST NOT wrap around the electro motor’s own axis as depicted here: Please Register or Log in to view the hidden image! That is it? So, if my goal is not to let the cable wrap around the axis the violet rod cannot help me? The angular speed will quite low, let’s say 0.1 rotation/sec, but the cable will be very long. Can it be calculated? At least approximately RJBeery Angular velocity at first will be zero (electro motor does not operate), then it will gradually (quite slowly) accelerate to some certain value Please Register or Log in to view the hidden image! This will happen in space, there will be no air Yes, I also think so, because the centrifugal force will work in this case I will paraphrase my question in a bit different way: at first the electro motor does not work, so the cable/tether is motionless and stretched. Then the motor begins rotate and “takes” the cable with it and thus making the cable to rotate. The question is: how quickly the signal, the information propagates through the cable that it (cable) should also rotate? How quickly each point of the cable will “feel” that it should obey the electro motor? I opened this topic because I need to answer this question that raised to some certain persons (they are scientists) and they asked me: “Why would an electric motor set a flexible tether in rotation? Would it not rather simply wind it up like a winch?” So, I need somehow (from the Physic’s point of view) show/prove that the cable when it begins rotating will not wrap around the electro motor’s own axis….
Oh typical. Pick something really flexible. Please Register or Log in to view the hidden image! Do you mean while it's running or is that condition you're applying for when the motor stops? Not if the motor stops suddenly (slowing down gradually should prevent it happening). But even with the violet rod if the motor stops suddenly then the free end will continue to rotate for a while. And WILL wrap the cable round the axle. That sounds like there will be significant lag. But that's just a gut feeling. I'll be utterly honest: I can't imagine there is no way to calculate it, but for the life of me I can't think how at the moment. It's not a problem I've come across in any of my work (so I've never had to use/ remember it) and college was more than 30 years ago. I'll try, over the weekend, to see if anything looks appropriate by Googling. I imagine I'd have a reasonable chance at recognising a potential solution on a quick skim of anything I can find. Edit: I just noticed this: I see their point. I had (mistakenly) assumed that the free end would be given an impulse to get it moving and the motor was more of a sustainer. You have problems with an elastic cable and the motor as the sole driver. But I'll keep looking and thinking.
I see...I think the answer is going to be a function of 2 ratios... 1) Length of cable / diameter of motor's shaft (or alternatively, length of cable / length of the violet rod being considered) 2) Rigidity of the cable / the rate of angular acceleration In the picture you have above my instinct tells me that it would wrap around the shaft at any rate of acceleration given a rope-like cable with very little rigidity. Sorry if this means you lose a bet!
I think it can be done. The violet rod will help, but not absolutely necessary. Proving it seems to involve vector calculus that is beyond me. This is what I've got. It could probably be greatly simplified, but not by me. I think my notation is correct, but if something doesn't seem to make sense then I've probably made a mistake. Assumptions: The cable mass is negligible compared to the mass attached to the free end. The mass attached to the free end is negligible compared to the mass of the motor. The cable is very flexible, and has a high tensile strength and modulus of elasticity, such that dynamic effects within the cable can be ignored. A limit on acceleration is included in the equations to ensure this is reasonable. (edit to add) The cable and motor system is in freefall and in vaccuum Variables: \(\vec{x}\) is the displacement vector from the centre of the spindle to the free end of the cable. \(\vec{y}\) is the displacement vector from the centre of the spindle to the attached end of the cable. Initial conditions: The cable is pointing straight out from the spindle: \(\vec{x}\cdot \vec{y} = \left\| \vec{x} \right \| \left\| \vec{y} \right \|\) ...or equivalently... \(\vec{x} = k\vec{y}\) Where k is a scalar constant, equal to c/r (defined below). The free end of the cable is not moving: \(\frac{d\vec{x}}{dt} = \vec{0}\) Dynamic conditions: The flexible cable is attached at a distance r from the center of the spindle: \(\left \| \vec{y} \right \| = r\) The flexible cable is always straight and taut, and always of length c: \(\left\| \vec{x} - \vec{y} \right \| = L\) Any acceleration of the free end of the cable is directly toward the attached end: \(\frac{d^2\vec{x}}{dt^2}\cdot(\vec{y}-\vec{x}) = \left \|\frac{d^2\vec{x}}{dt^2}\right \|\left \|\vec{y}-\vec{x}\right \|\) This acceleration must be kept below some arbitrary value to avoid dynamic cable effects: \(\frac{d^2\vec{x}}{dt^2}\cdot(\vec{y}-\vec{x}) < a_{max}\) The angle at the attached end of the cable between the cable and the direction of the spindle centre can never be acute (this is to ensure that the cable does not wrap around the spindle): \(\vec{y} \cdot (\vec{x}-\vec{y}) > 0\) Terminal conditions: Both the fixed end and the free end of the cable are rotating around the spindle at a constant angular velocity. This condition may be approached asymptotically. The problem is to find some function for \(\vec{x}\) over time that satisfies all conditions.
It is not clear to me what supports the cable and sphere when the motor starts from rest and speeds up. This is when problems will be likely to occur (or when the motor slows down to rest). While the motor is turning at constant speed, there will be no bending of the cable in the horizontal plane. It will hang downwards to some extent due to gravity, though. If the cable extends from the axis as the motor speeds up, there should be no problem. When the motor is turning at constant speed the cable will always be straight.
The way I see it (it's 5 AM here so mostly I see it through haze of "wish I'd gone to bed") is: if the cable is not rigid then how can the free end start moving at the same time as the motor starts up? Surely there will be delay between the two - and this delay will provide the opportunity for the cable to start wrapping... Which will simply exacerbate the problem. No? James' solution would be the one I'd pick from a practical engineering point of view, and had occurred to me just before I logged in.
No. The cable doesn't start wrapping unless it's at a tangent to the spindle. So there's a quarter turn of rotation to play with to get things going. This isn't much, since the only way to influence the free end is through the cable tension, so there is very little lateral force that can be applied. But, very little is not zero. I suspect it will always be enough, but the rotation will need to start arbitrarily slowly as the spindle radius decreases.
I see what you mean. (I was classing ANY movement that shifted the angle off directly radial as wrapping). So it's a conflict between the inertia of the free end mass and the acceleration of the spindle up to full rotation speed. If that's worded how I mean it. Aaagh. I'm going to bed. Thanks Pete. There is another for you if you've got the time: I think it boils down to "real world" vs. "ideal world" world physics - here. From post 15 - the question about cars turning the Earth underneath them.
I'm now thinking that there might be a minimum spindle/rod radius that would work. But, I really don't have anything concrete to deny or refute that, just vague handwaving ideas.
This question seems to be much more complex than I imagined…..Please Register or Log in to view the hidden image! Dywyddyr I do not care what will happen after the motor stops Please Register or Log in to view the hidden image! I need just to clarify if the cable/tether wraps around the red axis or not AFTER the cable is stretched and motor begins rotating….. Let it be! It is acceptable, my goal is to show that the cable WILL NOT wrap around the axis. Actually I think that this is clear from the daily logic. I think that all of us has taken the long rope (with stone attached at the rope’s end) in his hands and then rotate it. Have you seen that the rope to wrap around our body? I have not seen that……so, presumably the same will happen is space-when the cable is already stretched it probably will not wrap around the motor…. The free end will have impulse when the cable will begin deploying-as I wrote the rockets will take this free end (and sphere) to space. Of course it will be linear impulse and not circular (because the cable should be like straight line when deploying). Pete Well, I do not know this…it depends which material will be used for manufacturing the cable…so-difference between their masses is very important and should be taken into consideration? I think YES Yes, the cable MUST be made of very flexible and elastic material (like rubber) because at first it will be folded (something like yarn) and then it will be deployed. This is necessary? What if they are placed on the Moon? No. At first this velocity is zero (electro motor does not work) and then it gradually increase to some certain velocity. After this, yes, it may have constant angular velocity. So…and how? James R Simply imagine that this problem does not exist. Pete No, on the Moon……… Dywyddyr Very smart and clever idea! Indeed, the signal cannot propagate faster than light! So, it seems to me that if the rod is very long (and this is so) the electro motor should pick this (gradually increasing) rotational speed very slowly and probably exactly this circumstance excludes the chance of wrapping the cable around the motor’s axis…………………… Pete Well…..could you please somehow depict what you said? How the cable can (not) be at the spindle’s (you mean motor’s) tangent?
Hmm, consider a cotton bobbin with a length of cotton projecting from it, at the end of which there is a weight. If you spin the bobbin slowly will the cotton wrap or not? I think that when we spin a rock on string we "jerk" the stone into motion (I don't have string and stone handy - nor the room to try it!). Would that do the trick? A high-speed impulse and then spin at a reduced rate? Edit: or is it something to do with the initial impulse being provided off-centre of the eventual spinning? In which case the violet rod would help. Not quite the same thing as I meant. Strange. I see this a reason why it should wrap. Please Register or Log in to view the hidden image! No, look at your setup from above: the cable is radial to the spindle. The upper view is "at rest", the second is starting the rotation of the spindle (30[sup]o[/sup]), the third is the spindle even more rotated (another 30[sup]o[/sup]). The cable is getting closer to tangential with each increment of rotation of the spindle. Please Register or Log in to view the hidden image! Note: for ease I've rotated the cable/ mass rather than the spindle since it still shows the principle and keeps the diagram in-line.
Pete, I'm not quite sure the following are compatible If we must use dimensions similar to what Eagle posted, I believe there are only two possibilities: 1) Ensure the cable is rigid enough such that its minimum circumference of flex is greater than the circumference of the motor spindle. With a perfectly inelastic cable, this will spiral but not technically be "wrapped" around the spindle. Also, if it matters, after getting up to speed one could reduce the angular velocity by a marginal amount which would allow the end mass momentum to straighten out the cable again. 2) Ensure the cable has some sort of elasticity to it, such that its natural resting state tends toward being straight. That way the whole problem comes down to calculating a sufficiently slow acceleration rate. Without either of the above I maintain that avoiding wrapping is not possible with dimensions anything near the pictures posted. Consider spinning a yo-yo around your head...you do it by holding your arm out stiff. Now try to imagine doing that if the yo-yo was 10 meters long (it won't work). This is why I wrote that one must consider the ratio:
Careful... Your goal should be to determine the truth. We know that the cable will wrap around the axis under some circumstances. Our goal is to determine whether it is possible to run the motor such that the cable will not wrap around the axis. Have you tried that with a very long rope? It's not that easy. Problem solved. Begin rotation while deploying. You wouldn't need rockets then, because the sphere will be carried out by centrifugal force. That makes the problem harder, so that I can not mathematically describe it. Ah... so when you say elastic, you mean stretchy. There is confusion here, because elastic means something different in a technical context. Technically, elastic means that it will tend to return to its original shape or size when deformed. Something with a high modulus of elasticity is not very stretchy, because it strongly tends to retain its size or shape. A very stretchy cable makes a more difficult problem that I can't mathematically describe. Then James's solution works fine. Let it hang straight down to begin. It will swing out as the rotation starts. Don't know! I'm just defining the problem more formally.
These two are compatible under the assumptions that the cable is strong and resistant to stretching, (ie has a high linear modulus of elasticity and a high yield strength), and that acceleration applied at the fixed end is small enough that the propagation delay through the cable can be ignored. The implication is that the cable must always be under tension, ie the fixed end should always be under acceleration in a direction away from the free end. I don't know if this is compatible with the others or not, at least not for all values of c, r, and a. That's essentially the problem to be solved. Definitely. That was an astute observation. If there is a minimum ratio, and if the model I proposed is good, then this would mean there is a minimum value for c/r that allows a solution.
Dywyddyr If the bobbin has not got additional lever (I mean that violet rod) and if the cable is directly fastened on it, then I think that that cable will be wrapped around the bobbin Please Register or Log in to view the hidden image! That picture ensures me that the cable without that violet rod has got more chance to wrap around the motor’s shaft. RJBeery The cable should be very thin (otherwise it will too heavy!) and quite elastic (because at first it will be folded in box and then deployed)-just like Space Elevator’s cable that will be made of Carbon nanotubes. Well, cable’s length will be MUCH more that shaft’s diameter. As for comparison length of cable / length of the violet rod-again, the cable will be much longer. However this latter comparison will be less because it is possible to make the rod with the length more than the shaft’s diameter. Pete Well, I agree. No, and I understand that with very long it will be different….. During deploying the cable is folded and if I rotate the motor then cable will probably wrap around it….the fact that the cable will be stretched (I will explain this circumstance in my post’s end) will not enable the cable to “shorten”, to move towards motor and therefore (probably) the cable will not wrap around the shaft…. Of course, the rockets will not need to cover whole distance in space (cable’s length) but some part of it still should be covered. Oh, I did not know this….. Well I explain what I mean; otherwise it really confuses all of us: this cable will be (almost) the same as the cable for the Space Elevator. This cable (that will be made of Carbon nanotubes) will be very thin and long, you will be able to fold it in box, to roll/wrap around the bobbin and etc. But you cannot stretch it! I mean that if its length is 1 meter it will always have this length and never more or less (of course you may actually cut it but it is different matter). So, “my” cable WILL NOT be like rubber that you can really stretch, for example if there is cable/ribbon/tether made of rubber and if its length is 1 meter then you can stretch it and its length will be 5 meters, 10 meters and etc. But in our case the cable cannot be stretched in this way (we simply do not need it), so the word “elastic” I used in the sense that you can change its shape/direction (as if it is an ordinary rope, can you make it longer? No! But you can wrap it around any object) like this: Please Register or Log in to view the hidden image! This cable at first (left position) is straight, then it was bent but it does not mean that its length became more-this cable is NOT made of rubber….. I will do it, perhaps it was my mistake when I did not give enough data about this situation. So: this electro motor is based on the Moon, more precisely on its near side’s centre, somewhere at equator. The rocket engines fire and take the cable towards the Sun, the length of the cable is equal to several million kilometers (or maybe more!). So-when you ask me what makes the cable stretched (in the sense that it will be in straight position like arrow) I will answer-the Sun’s gravity will because due to cable great length cable’s major part will be in Sun’s sphere of gravity-beyond Hill sphere (or beyond L1 Lagrangian point, by the way I do not know at which point the Sun’s gravity prevails over Earth’s one but this is not important now). You may look at this situation: Please Register or Log in to view the hidden image!
People! Did I write something wrong in my previous post? Please Register or Log in to view the hidden image!
