Locking 2 turning objects - the inertia problem

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Firstly, let's assume that we have 2 perfectly identical objects. One of them is turning at 20 rad/s, but the other one - 10 rad/s. Now they get locked together. Let's assume no velocity loss during locking. What velocity will they each turn after being locked? It's pretty obvious that they'll be turning at 15 rad/s. Problem solved!

However, now let's make them both be solid cylinders with identical shape, but one of them be twice denser than first one. One of them is 5 kg heavy, but other one is 10 kg heavy, as example. Both of them got radius of 0.1 metre.

So inertia of the first one is calculated like so:

I = (5 * 0.1^2) / 2 = 0.025

For the second one it's:

I = (10 * 0.1^2) / 2 = 0.05

So the second one also got twice bigger inertia.

Now let's assume the same case. First cylinder is turning at 20 rad/s and second one is turning at 10 rad/s. They get locked. And now I get lost. I know well that they're gone end up turning slower than 15 rad/s as there is more momentum at the slowest turning cylinder. However, I don't know how to calculate how fast will they turn. So what I ask for is the calculation for this specific problem.

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It's pretty obvious that they'll be turning at 15 rad/s

Why is that so obvious ?

This is not. Are these objects turning around their gravity center ? their geometry center ? What makes or had made them turn ? Any kept forces ? Are these objects in the void ? On a perfect ice maybe ? Or are they on a rough ground ? What about their surface ? And about their weight and the gravity in which they might be ?

What I want to say is that without a perfect and complete definition of your system, there will be no relevant answer to your question.

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Conservation of angular momentum dictates that the sum of angular momentums before and after the locking should be equal: https://en.wikipedia.org/wiki/Angular_momentum . That leads to the answer.

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This requires no external torques. And the result will still depend on the points of interest, which we know nothing about.

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It's pretty obvious that they'll be turning at 15 rad/s

Why is that so obvious ?

This is not. Are these objects turning around their gravity center ? their geometry center ? What makes or had made them turn ? Any kept forces ? Are these objects in the void ? On a perfect ice maybe ? Or are they on a rough ground ? What about their surface ? And about their weight and the gravity in which they might be ?

What I want to say is that without a perfect and complete definition of your system, there will be no relevant answer to your question.

I already told that we assume no friction or such external forces and thus it's fully conserving angular velocities. Assume objects turning in void.

Conservation of angular momentum dictates that the sum of angular momentums before and after the locking should be equal: https://en.wikipedia.org/wiki/Angular_momentum . That leads to the answer.

So...

L1 = 0.025 * 10 = 0.25

L2 = 0.05 * 20 = 1

L = L1 + L2 = 1.25

And now... How do I split this momentum between both?

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And now... How do I split this momentum between both?

Proportionally to their moments of inertia.

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And now... How do I split this momentum between both?

Proportionally to their moments of inertia.

Oh, wait... Is it 1.25 / (0.025 + 0.05) = 16.6666667? If so, than it seems pretty much beliveable. At this specific case the momentum is conserved, but it's also turning faster than 15. I think that I just got how it works. Thanks!:)

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So what I ask for is the calculation for this specific problem.

that should be conservation of angular momemtum.

Alvaro be me to it! <g>.

Edited by Norman Barrows

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