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Having trouble grasping the Cross Product

  1. May 13, 2008 #1
    I know how to compute it and how to use it in equations. I do not understand what is physically happening however. When you turn a disk to the right the angular momentum is the cross product of the forces so it goes perpendicular to them and away from you. To the left would be towards you.

    But the disk isnt moving in those directions, how can it have momentum in the direction if it is just sitting there spinning? All the particles are moving around the disk and none are moving or even wanting to move in any perpendicular direction unless you apply another torque on the object giving it another cross product, then all of a sudden the two cross product forces start chasing each other and they come alive. I don't get it!!

    I guess to sum it up, what exactly is a cross product of two forces, not how do you compute it, but what is happening physically to the mass that involves the perpendiuclar direction?

    Also, why does right produce away and left produce towrds, who chose those directions, it is not a skrew threaded to do that.... Isnt space symetrical?
     
    Last edited: May 13, 2008
  2. jcsd
  3. May 13, 2008 #2

    tiny-tim

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    Hi DeepSeeded! :smile:

    This is just another application of good ol' Newton's second law F = mv.

    You're probably familiar and happy with the idea of dot-producting it with any vector k to give F.k = mv.k (the components in the k-direction).

    Well, if we cross-product it with any vector k, we get Fxk = mvxk. We call Fxk the torque in the k-direction, and we call mvck the angular momentum in the k-direction.

    "angular momentum" doesn't mean it's a momentum.

    It's just a name! :smile:
    Again … it's just a name … we call it "away", but we could equally turn all the names inside-out, and call it "towards".

    Like a ship is blown South by a North wind.

    We could call it a South wind, just to be consistent … but it would still be the same wind! :smile:
     
  4. May 13, 2008 #3
    I have only heard of F=ma nd F=mdv/dx :confused: but all the same...

    However the angular momentum does not point in the k-direction.. It points perpendiuclar to both forces. Thats whats got me, cross products do not have anything in common with either of the starting forces, it is out in its own dimension Eeek.
     
  5. May 13, 2008 #4

    rcgldr

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    Angular velocity vector = (radius vector x velocity vector) / (radius ^2)
    http://en.wikipedia.org/wiki/Angular_velocity

    Angular momentum vector = radius vector x linear momentum vector
    http://en.wikipedia.org/wiki/Angular_momentum

    Torque vector = radius vector x force vector
    http://en.wikipedia.org/wiki/Torque

    I don't know what the cross product of two forces is.

    Right hand rule is just a convention. I'm not sure if a charged particle's movment through a magnetic field follows right hand rule because of physics or because of convention for what is considered the "direction" of the field.

    Representing angular velocity, momentum, and torque as vectors acting along the axis of rotation or force is also a convention, but I'm not sure there are any good alternatives.
     
  6. May 13, 2008 #5

    tiny-tim

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    oops! :redface:
    hmm … I should have drawn myself a diagram … that's completely wrong … should be:

    Well, if we cross-product it with any vector k, we get Fxk = m(vxk)'. We call Fxk the torque, and we call mvxk the angular momentum.
    Well, it's perpendicular to the velocity vector v and a position vector k.

    For a solid body, which must rotate about an axis, we can combine vxk for opposite pairs of points, giving, for each pair, vx(k+a) = (-v)x(k-a), = 2vxa, so the position vector k disappears.

    So, for a single particle, the angular momentum is vxk, which depends on the observer, but for a solid body k drops out, and the angular momentum is independent of the observer.

    But it's still a cross-product, and so it must be perpendicular to the velocities of all the points in the body … in other words, it must point along the axis of rotation. :smile:
     
  7. May 13, 2008 #6
    So simply because it is a cross product it must be perpendicular. Which I do accept. But it doesnt really mean anything? There is no force or action in the resulting direction? Yet it is so useful to use the cross product of forces to know what direction a body under rotation will move... :bugeye:
     
  8. May 14, 2008 #7

    tiny-tim

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    Hi DeepSeeded! :smile:

    Yes … no force or action in the resulting direction … but there is a cross-product of force in the resulting direction. :smile:

    (It's ok if we take the cross-product of both sides of any equation.)

    Force is a vector, in a vector space, and that means more than just a line with an arrow on it.

    A vector in a vector space has several other properties that can be squeezed out of it.

    Taking the cross-product is just one way of doing that. :smile:
     
  9. May 14, 2008 #8

    Andy Resnick

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    Angular momentum and torque are not the cross product of two *forces*. L = r x p and t = r x F, where r is the moment arm. They do point perpendicular to the moment arm, along the axis of rotation, and thus describe why a spinning object precesses when a torque is applied.

    As for why the 'right hand rule' applies rather than the 'left hand rule', I suspect it's because the people that invented all this stuff were right-handed. The right hand rule also applies to how coordinate axes are oriented, leading to 'proper transformations' having a determinant of 1 (improper transformations have a determinant -1 and include an inversion)
     
  10. May 14, 2008 #9
    Correct me if I am wrong, but is r not a force? It must be a force of graivty or electromagnetic force in an object that must pull on the mass to keep it from leaving orbit.

    For an example in a spinning disk for each particle of matter, r (the distance from the center) is the force of the particles before it holding that peice of matter in orbit. Pulling on that peice of matter with the electromagnetic forces of the tightly bound atoms in the material.

    The greater r is the more force must be applied to hold each peice of mass in orbit because their angular velocity is higher and higher. r in a rotating object is just a measures of the force and is proportional to the distance.

    So that would mean that in both cases of L and t you are taking the cross product of two FORCES.

    I guess p is not a force, though it is still an object moving in a direction that has no relation with the vector produced by the cross product.
     
    Last edited: May 14, 2008
  11. May 14, 2008 #10
    No, r in L = r x p and t = r x F is not a force. It's true that in order to keep an object spinning there needs to be a centripetal force, but it turns out that this force does not change the rotation of the object. This is accurately modeled by the cross product t = r x F. Any force in the same direction as the radius vector will yield a cross product of 0, thus no torque and no change in rotation. In fact, using centripetal force instead of the radius vector in calculating L would lead to wrong results, since r is in general not equal to centripetal force in SI units.

    I am not as mathematically advanced as many people on this forum, so I do not know if there is some deep physical significance of the cross product. It seems to me, however, that cross products were not something we discovered in nature. Someone invented the cross product, and it turned out to have nice mathematical properties that modeled what happens experimentally. You are right that in the case of rotation, the direction of L or t may be arbitrary--I would just treat this as an artifact of the definition of a cross product. As Andy mentioned, the right hand rule is just convention.

    The direction of the cross product does become important in magnetism, but only because of conventional definitions of the direction of the magnetic field, etc. Yet, you'll find the convention useful when thinking about those problems.
     
  12. May 14, 2008 #11
    I do find it useful as the cross product of L always follows the cross product of any t. I just wish I knew what the connection between r, p, and the resulting direction of the cross product was physically, not just as a bunch of symbols :uhh:

    Sometimes math is just like a giant curtain over what I want to see :frown:
     
  13. May 21, 2008 #12
    I was reading about Magnetic Fields and it says that the direction of the magnetic force generated from a current is found by the right hand rule of the cross product. Great this cross product comes up again!!

    There was no explanation of why this is so. But this time it is a real force. So the cross product directions can't just be made up. Why does everyone just say to follow the right hand rule without an explanation?!?

    If the direction of electrons in motion is creating a specific perpendicular force this is proof against space symmetry.
     
  14. May 21, 2008 #13
    You're objections are well made! There is nothing physical going on perpendicular to a spinning platter--there is no perpendicular momentum or displacement from the center axis.

    As to another objection you raised, torque could just as easily been have been defined in the sense [tex]Z=Y\times X[/tex] as [tex]Z=X\times Y[/tex].

    But the cross product is a useful operator. Torque, for instance, is more than a magnitude. Say you have a vertically oriented screw you tighten by applying horizontal force in the X direction, displaced by a distance Y from the center of the screw. There would be a very different effect if you applied a vertical force with the same displacement!

    Some sort of vector--or three numbers, at least--are needed to account for torque. It turns out that calling the cross product a vector is something of a fiction, but a very useful one. You can rotate your coordinate system and it changes just as a vector does.

    Imagine for a moment we lived in 4 dimensional space (X,Y,Z,W). Then does [tex]X\times Y=Z[/tex] or does [tex]X\times Y=W[/tex]? Take you're pick, I guess. The cross product works because space is three dimensional. Two dimensions of space, taken two at a time, is three.

    In four dimensions there are six ways to take two dimensions at a time. This is a useful quantity too, but taking two dimensions at a time is not called a cross product in anything but three dimensions.
     
    Last edited: May 21, 2008
  15. May 21, 2008 #14
    say you have 2 vectors X and Y which lie on a plane and are subtending an angle theta between them. then the cross product, XxY, is physically equivalent to taking X and rotating it by an angle theta so that it is colinear with Y. the resultant is perpendicular because it identifies the axis of rotation of X.
    ditto with torque. when you take the cross product rxF, the resultant vector which you get is actually identifying the magnitude of the torque and the direction of the axis of the rotaion caused by the torque.
     
  16. May 21, 2008 #15
    The cross product is defined, this definition does not change... The vectors used in the cross product can change, depending on what those vectors represent they change the meaning and understanding of what the cross product resultant vector is representing. In the torque case the cross products length is representing the magnitude of the torque and the direction of rotation. You must understand though, that it is the meaning of the vector r and the meaning of vector F that gives the cross product its meaning of rotation in this case. To interpret the direction of rotation you would align your right hands thumb in the direction the vector points and rotation is in the direction your fingers curl... this is really just to aid the human mind in remembering what direction the rotation is in when you look at this type of vector...Just as the right hand rule helps the brain identify which direction the cross product should point in...

    ... If you can think of a better way of representing rotation in 3D space let me know.
     
    Last edited: May 21, 2008
  17. May 21, 2008 #16
    There are one or two, all equivalent of course, involving grassmann algebra.
     
  18. May 22, 2008 #17

    tiny-tim

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    Hi DeepSeeded! :smile:

    Think of the cross-product as the normal to what you're interested in.

    Have you done lines and planes in three-dimensional coordinate geometry yet?

    It's a lot easier to define a plane by defining its normal (any line perpendicular to the plane) than by defining the plane itself! :tongue2:

    Similarly, if a disc is spinning so that various bits of it are moving in the x-direction, and in the y-direction, and in all the directions in between …

    … it's a lot easier to define the normal to those different motions than to try to define them directly! :smile:
    No, it's not a fiction … the cross-product really is a vector (strictly, a pseudovector, since it goes all funny if you reflect it in a mirror).

    Cross-products can be added to each other in exactly the same way as vectors.

    In other words, cross-products obey the law of vector addition.

    (But don't try adding cross-products to ordinary vectors, since as I said , technically they're pseudovectors.)
    No … it's a proof against electron symmetry.

    If an electron follows the right-hand rule, then a positron (identical to an electron, but with opposite charge) follows the left-hand rule.

    Saying "this has negative charge" is simply a convenient way of saying "this follows the right-hand rule".

    Some things do, some don't … "positive" and "negative" charge are simply way of describing which is which! :smile:
     
  19. May 22, 2008 #18
    No matter how you cut it, simply providing a descriptive mathematical model, where the thing is defined from the beginning to accept as axiom that the result will be at right angles to the input vectors (cross product) does not help (me at least) get down to what is going on.

    Cross product may do an admirable job in aiding calculation of the resulting forces. What needs answering is why they act the way they do. If this can be contrived by proving that the incremental masses that make up a spinning disk would inevitably do this, that would be great. Maybe someone has, and maybe I can't quite see how to do it myself, but it would be very satisfying to get to understand. Is it even possible the Cross-Product definition was originally motivated because of the commonly observed phenomena?

    I wish I had read this thread before posing another related one (Why Right-Angles?)
     
  20. May 22, 2008 #19

    tiny-tim

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    Hi GTrax! :smile:

    Forces can be calucated this way because

    i) they obey Newton's second law, ∑F = ∑mv

    ii) F and v are vectors, and so they obey all the mathematical vector rules, including cross-product

    iii) so the cross-product Newton's second law with r, ∑r x F = ∑r x mv, must be true

    iv) so we take advantage of this by defining r x F as the torque, and r x mv as the moment of momentum (or angular momentum). :smile:
     
  21. May 22, 2008 #20
    I understand that it gives the axis of rotation, however the axis has two directions.

    If you turn a skrew to the right it goes away from your, if you turn it to the left it goes to you. This is because it is threaded this way.

    If an electron passes by and causes a magnetic field in only 1 perpendicular direction what makes it always choose the same direction and why does it follow the cross product rule.

    The cross product should be returning all possible perpendicular directions which would be 2.
     
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