Where is this guy getting his numbers

[skysunsand]

Homework Statement

I'm working through a practice test my professor has given us. The question is as follows:

Two uniform solid cylinders, each with radius R and mass M are spinning about their individual axes, with angular velocity w1, each in a counterclockwise direction. They are brought together so they rub against each other and eventually they have stuck together at one point. They are now rotating about the point where they have stuck together with angular velocity w2.

We are supposed to show that w2= 1/3 w1

Homework Equations

His work looks like this:

L1 = I1w1 + I1w1 = (1/2 MR^2 + 1/2 MR^2)w1 = MR^2 w1

L2 = 2 (3/2 MR^2) w2 = 3MR^2W2

The Attempt at a Solution

My question is where is he getting L2 from? Where is 3/2 coming from, why is he multiplying by two, and is there an easier or more intuitive way of solving this problem other than his solution?

 

[gneill]

Looks like an application of the Parallel Axis theorem. Where's the new axis of rotation for each cylinder with respect to their originals?

 

[skysunsand]

They've stuck together and are spinning like that.

I don't get how he's using the parallel axis theorem to get any of that.

I can se how maybe I1w1 + I1w1 = I2W2, because momentum is conserved, but where does parallel axis fit in there?


Attached is the picture.

 

[gneill]

Well, what's the moment of inertia of a cylinder (or disk) rotating about a point on its rim?

 

[skysunsand]

Well, what's the moment of inertia of a cylinder (or disk) rotating about a point on its rim?

I = 1/2 Mr^2.

So then putting that into either equation would give1/2 Mr^2 = 1/2 Mr^2 + Mr^2

and

1/2Mr^2 w1 + 1/2 Mr^2w1 = 1/2 Mr^2w2

 

[gneill]

I = 1/2 Mr^2.

No, that's its moment of inertia about its central axis. You want to move the rotation axis to the cylinders surface.

 

[skysunsand]

Then wouldn't it just be MR^2, because then it's a point mass?

 

[cepheid]

Then wouldn't it just be MR^2, because then it's a point mass?

How is it a point mass? It's still a cylinder. It's just rotating around a different axis than it was before. As has already been pointed out, you can use the parallel axis theorem here.

 

[skysunsand]

How is it a point mass? It's still a cylinder. It's just rotating around a different axis than it was before. As has already been pointed out, you can use the parallel axis theorem here.

I'm absolutely, hopelessly bad at physics. I have no clue how any of this is coming together, conceptually.


Here's all I understand:

I1w1 + I1w1 = I2W2

So supposedly, for some reason, you can plug in the parallel axis theorem, I guess to all of the I's to get-

(I+MR^2)w1 + (I+MR^2)w1 = (I+Mr^2) w2

Which still leaves me ridiculously confused, not sure how he got any of his numbers, and incredibly frustrated.

 

[gneill]

No. You use the parallel axis theorem to determine the moment of inertia of a cylinder rotating about an axis that is not its central axis. You only apply it to the expressions that concern the cylinders that are rotating about such an axis. Initially both are rotating about their individual central axes so you don't apply it there.

When the cylinders "stick" their new axis is parallel to the old ones but at the surface of the cylinders. Both cylinders rotate mutually about this new axis. Both will have moments of inertia about this new rotation axis that can be calculated using the Parallel Axis theorem. Find that moment of inertia. Sum them (there are two cylinders).

Conservation of angular momentum applies. Use it to determine the new rotation rate for the pair.

 

[skysunsand]

So that's where he gets L1- each of them are turning by themselves, so they're just

(1/2 MR^2 + 1/2 MR^2)w1.

But then okay, the second one is where you have to put them into parallel axis.

I= I + MR^2 is the equation for parallel axis. All I understand is that maybe you still put 1/2 MR^2 into one of those I's to get

I = 1/2MR^2 + MR^2

Which is where he got 3/2 MR^2!

And then, because it's more than one cylinder, it has to be multiplied by two. And then W2 gets tossed in there to prove w2 = 1/3 w1.

Frustrating, but finally. Thank you for your help. I've been trying to figure out physics problems for about 12 hours now and I don't feel like I've gotten any further from when I started. :/