# Why is angular motion special?

I'm a biochemist, and not a physicist so bear with me if I say something stupid. I do like to read about physics, though and this is one thing I can't wrap my head around.

I was recently reading about Mach's principle, and I realized that I don't understand what is so special about angular motion, and why it can't be considered a sum of a lot (infinite number) of smaller linear motions. If you consider a rotating ball as a whole, it is clearly rotating. But if I zoom in on any infinitesimal particle in the ball, it is not rotating but instead orbiting around the center of the ball. Orbits are well described as linear motion, with acceleration to the center in this case from whatever forces/chemical bonds are holding the ball together. The only particles that you might say are "rotating" are the ones lying exactly on the axis of rotation, but you could also say that these particles are just not moving.

Looking at rotations as an infinite number of orbits, I don't understand why we need to consider angular reference frames. Could a system of physical laws not be written that doesn't consider angular motion as a distinct thing from linear motion? This would seem to be a "simpler" system. Angular motion would be a tool to simplify complex problems rather than a true thing distinct from linear motion.

The only explanation I can think of is that you can't consider infinitesimal particles because they are quantized. Eventually you get to the smallest particle, which may be rotating around its center of mass. Maybe you could call this a "true rotation" but this is an unsatisfying answer to me. It is hard to believe that a large-scale physical concept (like the use of an absolute angular reference frame despite linear reference frames being relative) is dependent on the presence of quantized matter. Maybe its true, though.

Again, I might be horribly misinterpreting something, but it would be great to get an explanation for why we need angular motion to be something special and different from linear motion.

• For the purposes of this question, what is your definition of "linear motion"? – probably_someone Dec 8 '17 at 16:02
• I guess it would be velocity and/or acceleration with units of distance rather than units of rotation. – stords Dec 8 '17 at 16:06
• Ah, ok. The thing you're missing here, then, is that angular motion and "linear" motion are very easily interconvertible using the relation $v=r\omega$. They're not really distinct at all, it's just that it's usually more convenient to talk about the angular motion of rotating/revolving bodies rather than the tangential motion. – probably_someone Dec 8 '17 at 16:11
• Yeah sure, but angular momentum is not convertible I thought. It must be conserved independent from linear momentum – stords Dec 8 '17 at 16:15
• And this is true because the conservation laws are generated from symmetries of spacetime. Space looks the same under translation (i.e. lengths of things don't change when you move them around), so linear momentum is conserved. Space looks the same under rotation (i.e. lengths of things don't change when you change their orientation), so angular momentum is conserved. They must have two separate conservation laws because translation is fundamentally different than rotation. – probably_someone Dec 8 '17 at 16:18