Conservation of Angular Momentum is Dumbfounding

[fbs7]

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?

 

[russ_watters]

...while the total angular momentum around origin of the two marbles is zero (they cancel each other)...

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.

 

[fbs7]

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.

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:

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!

 

[Dale]

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)?

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.

 

[fbs7]

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.

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!

 

[Dale]

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!

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.

 

[Tom.G]

Such a cloud would not start spinning. It would just collapse into a central star which would not rotate.

Which brings up the question of why does our Sun rotate? (or equivalently, why was its generating gas cloud rotating?)

 

[Dale]

Which brings up the question of why does our Sun rotate? (or equivalently, why was its generating gas cloud rotating?)

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.

 

[fbs7]

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.

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!

 

[jbriggs444]

Just like a tank of completely immobile water, open the bottom, voila you got rotation.

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.

 

[Dale]

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!

The water is not completely immobile. At a minimum it is already rotating at a rate of about 1 revolution per day.

 

[fbs7]

The water is not completely immobile. At a minimum it is already rotating at a rate of about 1 revolution per day.

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?

 

[anorlunda]

no rotation

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.

 

[fbs7]

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.

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

 

[jbriggs444]

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

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.

 

[Dale]

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?

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.

 

[anorlunda]

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.

 

[fbs7]

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.

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?

 

[jbriggs444]

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.

You are trying to explain the result of an experiment that you have not run.

 

[A.T.]

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.

Have you tried spinning something on a string, and then reducing the length of the string?

 

[anorlunda]

dramatically

You are making a mountain out of a molehill. Take away words like "no" and "dramatically" and put instead "slight" and you should see there is no issue here.

The way natural language works, Alice can say star A rotates fast, Bob says star B rotates slowly. You and I can't be sure that A rotates faster than B.

 

[fbs7]

It's because I keep being answered that the reason why there's a whirlpool in a sink is that there is conservation of angular momentum on that fluid.

That is, if I leave my water undisturbed for a month in the sink then open the plug, the total angular momentum of the water while the whirlpool is happening is the same as the total angular momentum of the water before opening the plug, after it was left to rest for a month.

I read two explanations in this thread:
(a) There's an angular momentum in the water because it rotates around the Earth, so when the water moves into the plug conservation of angular momentum makes the whirlpool
(b) There are tiny movements in the water, so when the water moves into the plug conservation of angular momentum creates the whirlpool

Meanwhile I keep asking if it is the force of gravity what creates the whirlpool, not conservation of angular momentum, but I'm doing a bad job at asking that properly.

 

[jbriggs444]

I read two explanations in this thread:
(a) There's an angular momentum in the water because it rotates around the Earth, so when the water moves into the plug conservation of angular momentum makes the whirlpool
(b) There are tiny movements in the water, so when the water moves into the plug conservation of angular momentum creates the whirlpool

Both amount to the same explanation. The Earth's rotation is one reason that a tiny movement in the water would be expected.

Meanwhile I keep asking if it is the force of gravity what creates the whirlpool, not conservation of angular momentum, but I'm doing a bad job at asking that properly.

It is the reduction in radius resulting from the force of gravity. Conservation of momentum demands that tangential velocity and angular velocity increase.

 

[Dale]

It's because I keep being answered that the reason why there's a whirlpool in a sink is that there is conservation of angular momentum on that fluid.

That is, if I leave my water undisturbed for a month in the sink then open the plug, the total angular momentum of the water while the whirlpool is happening is the same as the total angular momentum of the water before opening the plug, after it was left to rest for a month.

I read two explanations in this thread:
(a) There's an angular momentum in the water because it rotates around the Earth, so when the water moves into the plug conservation of angular momentum makes the whirlpool
(b) There are tiny movements in the water, so when the water moves into the plug conservation of angular momentum creates the whirlpool

Meanwhile I keep asking if it is the force of gravity what creates the whirlpool, not conservation of angular momentum, but I'm doing a bad job at asking that properly.

Don’t forget (c) the container exerts a torque on the water

 

[fbs7]

I'm definitely dumbfounded - ergo the title of the thread :^)

If I put a needle on its point and my system is the needle and its surroundings, then any tiny imbalance in the needle will be amplified by the force of gravity (an external force), and that external force causes the imbalance to increase and the needle to fall. The fall of the needle is caused by gravity, and gravity increases the linear momentum of the needle, there's no conservation of linear momentum because there's an external force at work.

For my heart of hearts I thought it was the same about water in a sink (my system being the sink and its water): any tiny imbalance in the water would be amplified by the force of gravity (an external force), and that external force causes the rotation (and the water to go down the drain). The water going down and the drain and its rotation would be caused by gravity, and gravity increases the angular momentum of the water, there's no conservation of the angular momentum (or linear momentum either) because there would be an external force at work.

Yet it seems gravity causes the increase in linear momentum of the water in the sink, but does not cause the increase of the angular momentum of the same water. Science is really mysterious!

 

[Dale]

The water going down and the drain and its rotation would be caused by gravity, and gravity increases the angular momentum of the water, there's no conservation of the angular momentum (or linear momentum either) because there would be an external force at work.

That was the point of my (c) above. However, the direction of the angular momentum of a vortex is vertical, and gravity can only make horizontal torques. So any torque has to come from the container not from gravity.

 

[fbs7]

Huh! That's a good point! Gravity is vertical! Hmm... that's a conundrum, indeed

 

[fbs7]

Holly choo-choo! Just found me some equations of motion of marbles going down a funnel, and adapted (or tried to, haha) that to water. In cylindrical coordinates where the particles are following symmetrical paths around the origin, so that z = z(r) only

$$ (r, \phi, z), then\\
T = kinetic = \frac 1 2 mv^2 = \frac 1 2 m( \dot r ^2 + r^2 \dot \phi ^2 + \dot z^2 )\\
U = potential = m.g.z(r) $$

then the guy defines something called a Lagrangian (which I have no clue what it does, but I believe it does something) as L = T - U, and then he states that this is true:

$$ \frac d {dt} ( \frac {\partial L} {\partial \dot \phi} ) - \frac {\partial L} {\partial \phi} = 0 $$

then even I can calculate that

$$ \frac d {dt} ( \frac 1 2 m . 2r^2 . \dot \phi ) = \frac {\partial L} {\partial \phi} $$

Then the guy stated that L depends on r alone, not on φ, so "naturally" he said

$$ \frac {\partial L} {\partial \phi} = 0 \\
\frac d {dt} ( \frac 1 2 m . 2r^2 . \dot \phi ) = 0 \\
m.r.r. \dot \phi = constant \\
m.r. \omega = constant \\
angular momentum = constant $$

Hooray! I'm proud of myself for going from marbles to water down the drain! But, there were two assumptions done in that sequence:

(a) the trajectory of the particles is quite symmetrical around the origin, so that z = z(r) only, not z = z(r,φ)
(b) that Lagrangian thingie depends on r and φ' alone, not on φ

But there it is, proven by the equations, water going down the drain conserves angular momentum. I guess the same happens for gas collapsing into a star. Really fascinating and completely counter-intuitive!

 

[Bernard OByrne]

Does the Coriolis effect not explain the rotation of water leaving the sink?

 

[jbriggs444]

If you are talking about the Coriolis effect arising from adopting the rotating reference frame in which the Earth is at rest then no. In a household sink, other effects dominate. Coriolis is negligible by comparison.

 

[Buckethead]

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

Could you clarify this. Aren't the marbles going in opposite directions: i.e. 180 degrees apart? Why are you saying they have the same angular direction?

Are you saying that the momentum of the marbles is not zero because their trajectories draw parallel lines and not a single line (if the trajectory lines are extended in both directions)? Sorry for my thick brain on this one.

 

[russ_watters]

Could you clarify this. Aren't the marbles going in opposite directions: i.e. 180 degrees apart? Why are you saying they have the same angular direction?

If the stick is vertical and you fire a marble to the right from the top end and to the left from the bottom end, both are being fired clockwise (albeit on tangents), pushing the stick counterclockwise.

 

[Buckethead]

If the stick is vertical and you fire a marble to the right from the top end and to the left from the bottom end, both are being fired clockwise (albeit on tangents), pushing the stick counterclockwise.

I see. Interesting. So does this mean that if you have two marbles passing each other in space (parallel lines, not the same line) their total momentum is not zero?

 

[jbriggs444]

I see. Interesting. So does this mean that if you have two marbles passing each other in space (parallel lines, not the same line) their total momentum is not zero?

Total linear momentum can be zero. If it is zero (e.g. if you use a center-of-momentum reference frame) then the total angular momentum is guaranteed to be non-zero.

 

[Buckethead]

Total linear momentum can be zero. If it is zero (e.g. if you use a center-of-momentum reference frame) then the total angular momentum is guaranteed to be non-zero.

Got it! Thank you. I can see this as well if a line is drawn between two passing masses, the angle of the line changes as the distance between the masses increases.