# Conservation of angular momentum

We were learning about angular momentum in class today, and although it sort of makes sense, it's much harder for me to think about than linear momentum. So from what I can tell:

• Angular momentum and linear momentum are conserved separately: i.e. they can't be converted to one another to 'compensate' for an apparent drop in net momentum

• Every torque has an equal and opposite torque

Assuming those are true, consider the following scenario:

A disc with a rocket attached tangentially applies a thrust and thus a torque. The disc will go and fly in a circle, and ejects gas molecules in a circle, so linear momentum is balanced. The disc also has angular momentum, but the gas molecules all only go off in straight lines, so where is the torque opposite to that applied to the disc?

• Wait, so you propose that the disc will spin along its central axis? Does it have translational motion or is it constrained such that its center remains stationary? – codeAndStuff Oct 4 '13 at 18:25
• youtube.com/watch?v=NeXIV-wMVUk – mcodesmart Oct 4 '13 at 18:28
• It isn't constrained along its central axis, i.e. it is free floating in space. – user30575 Oct 4 '13 at 18:29
• why does there have to be a torque opposite? Wouldn't that result in static equilibrium? – codeAndStuff Oct 4 '13 at 18:35
• Also, are you assuming that the disc+rocket are somehow still a circular disc of uniform density? i.e. is the moment of inertia I= MR^2/2 ? Even so, as the rocket spends its fuel, this system will have a changing MOI presumably. This might not be as easy of a question as you want it to be from an introductory standpoint.. – codeAndStuff Oct 4 '13 at 18:41

The cloud of molecules emitted from the rocket has angular momentum. They move in straight lines but the lines do not have one single center.

We can observe the angular momentum of cloud of molecules by extending your experiment.

Lets place a big cylinder around the disk. The disk and the cylinder have the same axis. Cylinder has no weight and it is sticky so the molecules are captured on impact. The cylinder can spin around its axis.

What happens when molecule hits the cylinder?

• For a while lets assume the cylinder is not big - it is about the diameter of the disk - only one millimeter bigger. The molecule hits the cylinder almost tangentially. Molecule is captured and it rotates with the cylinder.
• Now imagine cylinder with bigger diameter. Impact of molecule is not completely tangential but still it makes the cylinder to rotate.
• We can enlarge the cylinder - the bigger it is the slower it rotates. But the impact of molecules will always make it rotate, because the direction of impact is not from the center.

So the torque opposite is the torque on the "cloud of gas molecules".