Quote:Original post by arithma
Quote:Original post by Way Walker
Look at it this way, if the body is at rest but rotating, then the sum of the linear momenta will be 0 by definition, so finding a formula for the linear momentum of each particle is not equivalent to finding the angular momentum.
Well am not proposing to find a relation between "sum of linear momentum" and "angular momentum". What am proposing is starting the reasoning from the (1) body being rigid and (2) the sum of linear momentum is zero. If these conditions must be satisfied, then the body MUST be rotating. Relying on this we MUST be able to find a formula for p given r (using same meanings for symbols). I believe if somebody may be able to find such a relation, it will provide instantaneous insight into angular momentum...
Given L and r, you can't find p. Cross product isn't reversible. You realized this when you pointed out that doubling r and halving p gives the same result. Doubling r and changing the angle from 90 to 30 would also give the same result. Cross product is lossy. You'll have to find some other source of insight.
Quote:
Moreover, you must understand that angular momentum is definetly not a concept by itself and apart from linear momentum. It is only a simplification. That emerges from the fact the Newton's Three Laws are enough to explain any phenomenon in classical mechanics; thus angular momentum is a derived concept. Yet we never see any literature concerning the "DERIVATION" of a concept. This was expectable at high school but I never imagined that it may prevail in the engineering physics courses. I don't know about physics majors: so is this really the situation?
Engineering is dirty physics in the same way that physics is dirty math. Really, engineering is about results and doesn't much care how the physicists came up with the results. In fact, you probably won't get to the derivation until upper level, even graduate pure-(as-opposed-to-dirty/engineering-)physics courses. Our classical mechanics course wasn't until junior/senior year.
If you really want it, here's how I see angular momentum coming about:
Consider an object in a force field (e.g. gravity, part of a rigid object, etc.). Let's say it's in orbit. It's orbiting so obviously momentum isn't conserved. However, it won't just spontaneously stop spinning, so something's being conserved. Energy? Maybe, but if its orbit contracts (think figure skater) that won't be conserved. The quantity (r x p) is conserved. This is somewhat intuitive if you picture an orbitting object; in an elliptical orbit v (and thus p) decreases with r. You don't escape this problem with a circular orbit because the direction of v changes, but (r x p) will always be normal to the orbitting plane. Basically, r "corrects for" the changes in p.
Now, I bet you have a couple concerns about what I just said:
1) "But momentum is conserved if you take into account what's causing the force field". True, but the same complaint could be brought against the claim that momentum is conserved even when energy isn't. Energy is conserved, if you take everything into account. Like you said, these are simplifications. Even Newton's three laws can be considered simplifications of General Relativity.
2) "But angular momentum is only conserved in the absence of non-central forces". Yeah, but this requirement is basically satisfied by requiring it to be in an orbit.
If you want something more rigorous, I think a book or pen and paper are your best bet.
This is going to sound critical, but it's not, I'm honestly curious: Did you demand a rigorous definition of limits before you were satisfied with derivatives?
By the way, you didn't play my game: describe momentum to me. [grin]