Question about n rotating parallel cylinders

[Prishon]

There are n vertical identical parallel identical cilinders rotating around their length axes with the same angular velocity. The are somehow fixed wrt to Earth and brought together (on a rail?). After the contact there is no slipping and the cilinders are coupled to their neighbor cilinders. It is easy to see the cilinders end up as follows:

For even n there is no rotation anymore. All cilinders are at rest.

For uneven n the cilinders end up with contrary velocities. If there are n cilinders in contact then the absolute value of their angular velocities will be 1/n of the initial value.

For example, 3 cilinders (assuming their mass and initial angular velocities are 1) end up with angular momentum 1/3 and kinetic energy of 1/9 of the initial value.

But how is the energy dissipated? 1/9 of the momentum and energy are left in the cilinders. The rest of the momentum has flown into Earth. But where has the energy gone. Not into the Earth(well a very small part). Has it gone by friction? But what if the cilinders were toothwheels?

 

[Halc]

If there are n identical cylinders and each has angular momentum of p, then the total angular momentum (be they apart or brought together) is n*p. That or external torque was applied, but you don't indicate that.

So if two of them are brought together, they'll form a sort of spinning snowmanular-cylinder shape (yes, that's a word now). The energy went away via friction/heat when they touched but the momentum must remain, so the joined figure must still rotate with the original combined angular momentum.

 

[kuruman]

If there are n identical cylinders and each has angular momentum of p, then the total angular momentum (be they apart or brought together) is n*p. That or external torque was applied, but you don't indicate that.

So if two of them are brought together, they'll form a sort of spinning snowmanular-cylinder shape (yes, that's a word now). The energy went away via friction/heat when they touched but the momentum must remain, so the joined figure must still rotate with the original combined angular momentum.

Can you provide a reference or link for "snowmanular"?

 

[Prishon]

Can you provide a reference or link for "snowmanular"?

The momentum of say 3 cilinders is not the same before and after. In the beginning they all 3 rotate say clocjwise (with omega say 1) and afterwards they rotate with omega 1/3, tbe middle one anti-clickwise, as it must be since they are oushed against one another.

 

[Prishon]

If there are n identical cylinders and each has angular momentum of p, then the total angular momentum (be they apart or brought together) is n*p. That or external torque was applied, but you don't indicate that.

So if two of them are brought together, they'll form a sort of spinning snowmanular-cylinder shape (yes, that's a word now). The energy went away via friction/heat when they touched but the momentum must remain, so the joined figure must still rotate with the original combined angular momentum.

They are pushed together in fixed position wrt to the Earth. Didn't I mention?

 

[Prishon]

Can you provide a reference or link for "snowmanular"?

This question comes forth if I search the net for that word...:)

 

[kuruman]

The momentum of say 3 cilinders is not the same before and after. In the beginning they all 3 rotate say clocjwise (with omega say 1) and afterwards they rotate with omega 1/3, tbe middle one anti-clickwise, as it must be since they are oushed against one another.

So you have three cylinders spinning about their axes that are brought together to touch and look from above like this OOO Correct?
Then because of friction, their surfaces will eventually stop moving relative to each other, correct?
While they are in contact, the angular momentum of the three-cylinder system about the axis of the middle cylinder is conserved because there is no external torque acting on this system about that axis, correct?
If all of the above is correct, what kind of motion do you think the three-cylinder system is going to undergo? Remember, no cylinder can spin independently about its own axis because friction has put a stop to that.

 

[Prishon]

So you have three cylinders spinning about their axes that are brought together to touch and look from above like this OOO Correct?
Then because of friction, their surfaces will eventually stop moving relative to each other, correct?
While they are in contact, the angular momentum of the three-cylinder system about the axis of the middle cylinder is conserved because there is no external torque acting on this system about that axis, correct?
If all of the above is correct, what kind of motion do you think the three-cylinder system is going to undergo? Remember, no cylinder can spin independently about its own axis because friction has put a stop to that.

Almost right! But the whole system can' t rotate. That system is fixed wrt to Earth. So momentum is tranferred to the Earth.

 

[Halc]

So you have three cylinders spinning about their axes that are brought together to touch and look from above like this OOO

See? That's a snowman shape, except rotated on its side. Extend it in 3D and it's a snowmanular cylinder.
Sheesh, you guys need to lighten up.

But the whole system can' t rotate. That system is fixed wrt to Earth. So momentum is tranferred to the Earth.

Oh. That wasn't clear from the OP.

 

[Prishon]

Three identical cilinders (with fixed axes wrt to Earth) are identically rotating. When brought together (say on a rail) they end up with 1/3 of their initial rotational velocity. The middle one obviously rotating contra the others. They end up with 1/9 of initial momentum and kinetic energy.

Four (or 6,8,10, etc.) end up with zero velocity.

Uneven numbers end up with 1/n velocity.

 

[Baluncore]

But where has the energy gone. Not into the Earth(well a very small part). Has it gone by friction? But what if the cilinders were toothwheels?

So long as the cylinders remain counter-rotating and in line, all is OK.
But during the crush, when any three get out of line and form a triangle, the teeth will mesh and those three gears will lock together, then groups of three will lock, and so on.

This is a similar problem to the over-speed failure of a flat, circular disk flywheel breaking into three equal 120° pie-slices. What is the rate of rotation increase of the fragments, each of which has a lower moment of inertia than the original wheel.

 

[Prishon]

So long as the cylinders remain counter-rotating and in line, all is OK.
But during the crush, when any three get out of line and form a triangle, the teeth will mesh and those three gears will lock together, then groups of three will lock, and so on.

This is a similar problem to the over-speed failure of a flat, circular disk flywheel breaking into three equal 120° pie-slices. What is the rate of rotation increase of the fragments, each of which has a lower moment of inertia than the original wheel.

But they stay in line always. Their axes are fixed wrt Earth (they can only be brought together on one line; they always stay in line, no trianglez).