Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Johann Mahne on 13/09/2011 07:45:58
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In many Physics and Math books you'll find a topic on the torque cross product.
Given by T=FXR. Newton Meters.
It is orthagonal to the direction of rotation.
None of the sources that i have read ever state that this is not a true vector. It is even used to calulate precession.
It has caused me to scratch my head. If it was true, you could'nt pedal a bicycle and keep it upright.
I stumbled across an obscure forum that stated that the direction of this force only indicates the direction of rotation, and cannot be taken as a true vector.
I agree with this forum, but when i wrote to a scientist hosting a physics web site he said the vector is real.
Who is right?
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Hi Johann,
Is your problem with torque as a vector the fact that if the torque points sideways out a bicycle wheel, that the wheel just rotates instead of moving sideways?
If so, then the solution is that a vector is a mathematical construct. It has a magnitude and direction and those two things allow it to model a variety of physical effects. When modeling a force, which is a push or a pull, it points in the direction of the push or pull, so you expect the object to move in that direction.
In the case of a rotation, how can you represent that rotation? You can tell me in words that the object is rotating either clockwise or counterclockwise, and you can tell me what axis it's rotating about. Mathematically, that can be represented by drawing a line segment along the axis of rotation. Then to designate clockwise or counterclockwise, you can put the arrowhead on one end or the other of that segment. What you've drawn is a vector, but it represents rotation about the axis, not a push or pull along the axis. That's why torque is a vector: it's a nice way to represent clockwise or counterclockwise rotation about an axis.
(To be technical, torque is a pseudovector. This is a technical term for how it's magnitude and direction change as you change coordinate systems. It doesn't have a lot to do with answering this question, but I bring it up for completeness.)
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Thanks JP,
I agree with you.
It does seem strange that this point is never brought up in so many books that I have read.
Same thing with angular momentum.
Don't you think that there should be a special notation for pseudo vectors?
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What books have you read on it? I seem to recall seeing it in at least some intro physics texts. I don't know if all of them address that point.
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Honestly, i have never seen pseudovectors being named on any physics website or in any physics books that i have read.
I'm talking about calculus books and physics books such as "physics for scientists and engineers".
I'm not a scientist, so it could be that you have been reading more advanced books than i am.
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Well, the pseudovector part isn't terribly important for using torque to do basic calculations, so it wouldn't surprise me if it wasn't in there, or was glossed over in a sentence or two.
The fact that the vector indicates the axis about which something will rotate rather than a direction the wheel is being shoved is very important.
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Well, the pseudovector part isn't terribly important for using torque to do basic calculations
You are right as far as mechanical engineering calcs go.
I've seen it used to calculate precession. Which to me is a bit dodgey.
I agree that the vector is required, but if you are a math student and asked to calculate it don't you think that
the first thing that will spring to mind is that it's a real vector? The same for angular momentum. You would think that a momentum is occuring in the direction of the axis.
All i'm saying is that angular momentum and the torque cross product should have some type of notation.
Spare a thought for the layman
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Torque is a pseudovector.
There are other ways of doing the calculation that don't involve pseudovectors and the cross product. I learned them because cross product is defined only in 3 and 7 physical dimensions. I wanted to do calculations in four dimensions, so no cross product.
The way that works in any number of dimensions is with what are called bivectors, trivectors, quadvectors, on up to n-vectors. Instead of having a vector along the axis of rotation, you have a bivector that defines the plane of rotation. Then multiply that by another vector to get a trivector. In 3D that's the end.
In higher dimensions you can multiply a bivector times a bivector to get a quadvector, and so forth.
So, why did they come up with this confusing pseudovector thing? It's because physicists at the time didn't care even know about dimensions other than 3. They didn't consider such things at all. So the did the simplest thing that worked.
Why don't we change to a better system? It is quite similar, more natural, and equally simple. It is because of tradition. The cross product is so established that there is no way to get rid of it. There is a vast body of literature that assumes that the reader knows it. Not to mention the vast number of scientists who work only in 3D and would see no value in changing. If the world DID change then it would still be necessary to learn the old way, so it would be necessary to learn two systems instead of one. There is not enough advantage to changing.
I'm working through all this now. Once I'm done I'll post it all to a web site for those who are interested and you may learn all about n-vectors and wedge products, which are what are used instead of cross products.
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This is the also the first I have heard of pseudo-vectors (http://en.wikipedia.org/wiki/Pseudovector).
So, why did they come up with this confusing pseudovector thing?
It seems that the difference between a true vector and a pseudo-vector has to do with how they would work in a mirror image; a true vector works the same way in its mirror-reflection, while a pseudo-vector is reversed in a mirror. Angular momentum & Magnetism from a coil of wire are examples of pseudo-vectors.
In high school, I learnt the "right-hand rule" for electromagnets; if you hold your hand next to the coil of wire, your thumb points in the direction of the magnetic field. But if you view this electromagnet (and hand) in a mirror, the current will flow in the opposite direction, and your right hand will now look like a left hand. But (if I understand it correctly) in a mirror, current still obeys the "right-hand rule"(?)
We can't actually see the flow of electrons within a motor, when viewed in a mirror. But in principle it should be possible to watch someone ride a bicycle or spin a top in a mirror.
It seems that a pseudo-vector is a kind of Bizarro-world (http://en.wikipedia.org/wiki/Bizarro#Publication_history) vector....
Edit: I'm not so sure if an electromagnet in a mirror (http://en.wikipedia.org/wiki/Pseudovector#/media/File:BIsAPseudovector.svg) obeys a right or left-hand rule?
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physicists at the time didn't care even know about dimensions other than 3. ... Once I'm done I'll post it all to a web site for those who are interested
I would love to see a video of you riding a 4-dimensional unicycle [:D]