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*From*: "John S. Denker" <jsd@AV8N.COM>*Date*: Mon, 14 Mar 2005 20:54:34 -0500

On 03/14/05 18:09, Daniel S. Price wrote:

A classic application of conservation of angular momentum, often cited

in physics texts, is the joining of two cylinders rotating about a

common axis. When the cylinders are allowed to meet "face-on", the

angular momentum of the system is conserved. When cylinders rotating

about parallel axes meet "edge-on", however, the angular momentum of the

system may not be conserved.

Says who?

Angular momentum always obeys a local conservation law.

Period. No exceptions. So figure out what you mean by

"the system" and account for any angular momentum that

flows across the boundary of the system.

In the latter case, imagine two cylinders, one initially rotating and

the other stationary. If they are gently brought into contact,

frictional force between the cylinders acts to slow the original

cylinder's rotation and induce rotation in the other cylinder. HRW

(fifth edition), chapter 12, question 49, is an example of this

situation, and in the problem the authors claim that angular momentum is

not conserved.

That's a totally idiotic thing for them to say.

I say angular momentum always obeys a local conservation

law. No exceptions.

If they mean that the angular momentum of the system is

not conserved, I do not see the source of the external torque (unless

the frictional force is responsible, though it would seem to be internal

to the system as it acts only between the cylinders).

I don't have the cited text in front of me. But I can

guess what the setup is. We have not just two cylinders,

but two cylinders on *axles* with *bearings*. To accelerate

one of the cylinders, it is necessary and sufficient to

have a _moment_, i.e. a pair of forces separated by a

lever-arm. It is sufficient and conventional to consider

one force applied to the rim and the other applied to

the axis. If the axle is anchored to something outside

your "system" boundary, as I suspect it is, then this is

a major pathway whereby angular momentum is transferred

across the boundary.

Another version of the question makes an interesting

exercise also: consider two cylinders *without* axles

and bearings, just floating around in interplanetary

space. When they come into contact, some angular

momentum will flow from the rotating one to the initially

nonrotating one. The angular momentum of the two-cylinder

system will be strictly conserved, since there is no

pathway for transferring it across the boundary. The

wrinkle is that some angular momentum will be lost from

the "spin" coordinates and transferred to the "orbit"

coordinates. That is, the two cylinders will pick up

some nontrivial center-of-mass velocity.

The former case is also discussed in HRW5, chapter 12, question 53; the

coupling of the cylinders "face-on" is said to maintain angular momentum

(though friction between the cylinders seems to be responsible for their

eventual rotation as a unit).

In that case, the rims are colocated and the axles are

colocated, so the moments will be equal-and-opposite.

There is nothing special about the "face on" geometry,

except that it implies that the axles are colocated.

Why are the two situations different in how conservation of angular

momentum is applied?

The statement that angular momentum obeys a local

conservation law is *not* different between the three

setups considered above. It is *always* true.

It is also independent of what you choose as your

origin of coordinates ... provided you choose one

and stick to it consistently. BTW if you happen to

be working in the CoM frame of an isolated system,

you don't even have to choose one; all the laws

are manifestly independent of the choice ... while

more generally, moving your origin from one place

to another shifts the total angular momentum by

MV /\ X, where X is the vector from the old origin

to the new one, and MV is the linear momentum of

the CoM, and of course /\ means wedge product.

As Joel R. pointed out, you can't choose your origin

to be centered on cylinder #1 and centered on cylinder

#2 simultaneously; you have to choose ... except in

the less-than-general coaxial case.

It would be a good exercise to work the non-coaxial

case choosing an origin centered on neither cylinder,

just to convince yourself that the formalism works,

independent of the choice of origin.

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**References**:**[Phys-L] conservation of angular momentum question***From:*"Daniel S. Price" <dprice@JEFFCO.K12.CO.US>

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