Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

B Conservation of Angular Momentum is Dumbfounding

  1. Dec 4, 2018 #1
    I find conservation of energy and linear momentum to be quite natural to understand, but I find conservation of angular momentum really, really tricky. Let me give two examples:

    (a) I call my system as a stick with identical springs at its ends, facing opposite directions, each spring is coiled with an identical marble. The whole system is at rest, and the origin of my system of coordinates is the center of the stick. Therefore at T=0s my total linear momentum and angular momentum are both zero. Then at T=1s the springs release, shooting out the marbles at opposite directions, and the stick starts to spin. The total linear momentum is still 0, but while the total angular momentum around origin of the two marbles is zero (they cancel each other), the stick is spinning and will have a non-zero angular momentum, so the total angular momentum of my system is now non-zero. There were no external forces or torques... I can't understand how there can be any conservation of angular momentum if I applied no external forces or torques to my system, and now the thing is spinning...

    (b) I call my system a slightly asymmetrical cloud of gas at rest around its center of gravity, with no rotations. Total angular momentum is zero. Gravity starts to do its thing, and the cloud starts to collapse to its center of gravity, but the thing is slightly asymmetrical, so the collapse is not perfect - it will start to have differences in speed, which, for being a gas, will eventually cause the thing to rotate. At some point the cloud of gas is collapsing and rotating towards its center of mass, until it forms an excellent rotating disk on its way to making a star. The direction of rotation will somehow depend on these initial irregularities, but whatever it is, now the system has a non-zero angular momentum. But there were no external forces, just internal force of gravity.

    In both cases total linear momentum was conserved and total energy was conserved, but angular momentum was not conserved even if there were no external forces or torque... what am I missing here?
     
  2. jcsd
  3. Dec 4, 2018 #2

    russ_watters

    User Avatar

    Staff: Mentor

    This is not correct. Both marbles are being fired in the same angular direction, so they have a combined non-zero angular momentum.

    I'll let someone else tackle the second.
     
  4. Dec 4, 2018 #3

    AAHH!! Now I see it!!! OMG, you folks are so bright! I have been thinking about this for a whole month and in 10 seconds you got the error in the argument!! That's awesome! I'm flabbergasted - thanks so much for the explanation!

    Now, if I can abuse your kind goodwill, what if it's just one marble, and the origin of my coordinate system is located exactly at the marble, like this:


    p1.png

    On that coordinate system the marble is flying away from the origin, so Lmarble = r.m.v = 0 because r = 0, as the marble will have a radial movement. Based on what you said I I'm guessing that maybe there will be two components of the angular momentum of the stick, so the matter would be to prove that the stick will also have Lstick=0, because it has a rotation and also a linear component going on, and I guess their angular moments cancel each other (in this coordinate system, that is)?

    I think I learned something! Yay! You're the best!
     
  5. Dec 4, 2018 #4

    Dale

    Staff: Mentor

    That is exactly correct. The center of mass is moving to the right, giving it an angular momentum going out of the page wrt the origin. The stick also has an angular momentum wrt its center of mass that goes into the page. They cancel.
     
  6. Dec 4, 2018 #5
    That's why I love reading this site. Thanks again!!! What an incredible thing!! One bullet going one way in a straight line, a stick moving the other way in a straight line and spinning around itself, yet whatever (I guess non-rotating) coordinate system I choose, the total momentum is zero!

    That's is really counter-intuitive: it's easy for me to imagine that two bodies spinning in opposite directions to have total momentum zero, but that one spinning and the other going straight, that total L is zero... that is mind-blowing! Yet, somehow, now that you guys said it, it does make sense.

    Now, if I would only understand how a zero-momentum collapsing gas cloud isolated in a distance region in the universe would all by itself make a spinning disk that, I assume, by some odd mechanism would still have total momentum zero... hmm... now that sounds like a really deep mystery!
     
  7. Dec 4, 2018 #6

    Dale

    Staff: Mentor

    Well, It isn’t a deep mystery, just a horrendously complicated calculation. Such a cloud would not start spinning. It would just collapse into a central star which would not rotate.
     
  8. Dec 5, 2018 #7

    Tom.G

    User Avatar
    Science Advisor

    Which brings up the question of why does our Sun rotate? (or equivalently, why was its generating gas cloud rotating?)
     
  9. Dec 5, 2018 #8

    Dale

    Staff: Mentor

    Why wouldn’t it be rotating? A random gas cloud would have several prameters, each of which would be a random variable: initial mass, initial velocity, initial angular momentum, etc. Each variable has some distribution and therefore some likelihood of starting in an approximately zero initial state or not.
     
  10. Dec 5, 2018 #9
    Now I'm extra-dumbfounded... I'd never imagine a non-rotating star, as I'd never imagine a non-rotating tornado! I thought every time a gas would be pulled into a point by a great force, creating a speed higher than some limit, it would start to rotate all by itself. Just like a tank of completely immobile water, open the bottom, voila you got rotation. In my mind the gas cloud rotated the same way a sink does!

    In my naivete on that stuff, I thought that if the gas cloud is big enough and generated several stars, then the effect of one star with momentum +L would imply that all other stars (and other rotating stuff) would necessarily have together a momentum -L. But then I couldn't make sense if the entire gas cloud becomes a single star. Or...

    ... or, now that I write it, this thought seems silly? I know that the entire universe does not rotate, so Luniverse = 0. But then I know that the solar system has Lsolar > 0. Therefore if universe = solar + rest, then Lsolar + Lrest = 0. So the rest of the universe needs to have a non-zero angular momentum, what sounds preposterous as the rest of the universe is random and should also add up to Lrest = 0, not to Lrest < 0. Arggh... angular momentum is so complicated!
     
  11. Dec 5, 2018 #10

    jbriggs444

    User Avatar
    Science Advisor

    The outcome of such an experiment depends critically on exactly how immobile that "completely immobile" water is. If it is indeed completely immobile and you completely avoid external influences, the expected result is that you do not get rotation.
     
  12. Dec 5, 2018 #11

    Dale

    Staff: Mentor

    The water is not completely immobile. At a minimum it is already rotating at a rate of about 1 revolution per day.
     
  13. Dec 5, 2018 #12
    Hmm... so if you have a spaceship accelerating at 1 gravity, that as far as we can measure it doesn't rotate, and you open a sink with water, there will be no rotation?
     
  14. Dec 5, 2018 #13

    anorlunda

    Staff: Mentor

    No is too strong a word. That demands perfection, which we seldom encounter in real life.

    In principle, you could balance a sewing needle on a table and have it sit there for eternity. Give it a try and let me know when you succeed.
     
  15. Dec 5, 2018 #14
    Hmm... hmm... I see. Say my sink is a square 10 cm wide, and there is 10cm of water in it; so it has 1,000cm3 of water. Then I wait 30 seconds, and I see no movement in the water.

    Now I open a 1cm hole at the bottom, and I see a spinning cylinder of water 2cm wide, 10cm tall. That's some 30 cm3 of water, spinning at the center at say 1 Hz. If the 1 Hz spinning of my 30cm3 of water was due to some angular momentum of the 1000cm3 reservoir, then the reservoir would have to spin at in the order of something like 1/30 Hz. But then I waited 30 seconds and I saw no movement.

    That's a perplexing thing. Maybe that cylinder starts spinning very slowly, and then there's some torque from gravity that increases the rotation? There must be an external force at work
     
  16. Dec 5, 2018 #15

    jbriggs444

    User Avatar
    Science Advisor

    Angular momentum at a fixed angular velocity scales not just with the mass, but also with the square of the radius. So you are missing at least two additional orders of magnitude in the analysis of your bathroom experiment.
     
  17. Dec 5, 2018 #16

    Dale

    Staff: Mentor

    Yes. If there is rotation then either the ship exerted a torque on the water (and there would be an equal and opposite torque on the ship) or the fact of rotation indicates that your measurement was not sufficiently precise.
     
  18. Dec 5, 2018 #17

    anorlunda

    Staff: Mentor

    Just a puff of air, a warm or cold heart source, or the vibration of someone moving in the spaceship could add a bit of momentum to the water.
     
  19. Dec 5, 2018 #18
    Hmmm... in that example of the needle standing on its point, the reason why the needle topples is that the thing is unstable. There's a force of gravity, from which any deviation from the vertical will cause a movement, which increases the deviation and increases the movement. I suspect that the increase of that movement is not due to a conservation of something, but by the fact that there is a force acting on it .

    Then you have that mass of water in a spaceship, and a force similar to gravity pulling it through a hole. So if it has a tiny unbalance of something, then the force similar to gravity will have some effect, and I suspect that will cause it to rotate. I can make my peace that the force will increase that unbalance through some mechanism, and eventually make a vortex in it, but I can't wrap my mind that the conservation of a microscopic unbalance of angular momentum alone will cause it to rotate so dramatically when it goes down the drain.

    For example, in that spaceship, if the acceleration similar to gravity is making the water to rotate "left" when it goes through a hole, then conservation should make the rest of the spaceship to rotate "right"; so I see the ship rotating right due to conservation of angular moment, but I can't see the water rotating left due to conservation of some microscopic amount of angular moment.

    Therefore, in the spaceship, isn't there any mechanism, like the one for that standing needle, that converts potential energy into kinetic energy in the water and amplifies tiny deviations to make them a big circulation?
     
  20. Dec 5, 2018 #19

    jbriggs444

    User Avatar
    Science Advisor

    You are trying to explain the result of an experiment that you have not run.
     
  21. Dec 5, 2018 #20

    A.T.

    User Avatar
    Science Advisor

    Have you tried spinning something on a string, and then reducing the length of the string?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted