Moving along through our countdown to Newton's birthday, we come to the next important physical quantity, angular momentum. For some obscure reason, this gets the symbol L, and the angular momentum for a single particle about some point A is given by:
This is probably the most deceptive equation we'll see this season. Yesterday's definition of work clearly showed its vector calculus roots, but to the untrained eye, this just looks like a simple multiplication: You take the momentum (p) and multiply by the distance (r) from point A, and you're all set.
To those with a little mathematical training, though, that × symbol is a terrifying sight. In math and physics circles, ordinary multiplication is so basic that it doesn't require a specific symbol. The × sign that ordinary people use to indicate multiplication is reserved for the most daunting of vector operations, the cross product. The linear momentum of the particle and the radius from point A are both vectors, having a direction as well as a magnitude. The cross product takes two vectors and produces a third vector-- thus, the little arrows over all three quantities.
What's more, the vector that results from the cross product is at right angles to the two vectors that were multiplied together to get it. So, if the momentum vector points north and the radius vector points west, the angular momentum vector points straight up. This is consistently the single most confusing thing in an introductory mechanics course, which makes it kind of a travesty that we only get to it in the last week or so of the term.
So, why do we bother with all this scary and confusing math?
Angular momentum is important because there is clearly something special about rotational motion. As anybody who has every played with a top or a gyroscope knows, spinning objects behave in slightly strange ways, to such a degree that every so often a crazy person will convince themselves that if they combine enough spinning objects in just the right way, they can defy the force of gravity. They can't. Angular momentum is weird, but it's not magic.
Angular momentum is the property we use to characterize the motion of things that move in closed loops, and it's one of the most significant properties in all of physics. On the largest scales, angular momentum is a huge factor in determining the structure and behavior of planetary systems, galaxies, and black holes. On the smallest scales, angular momentum is what determines the structure and chemical behavior of atoms. Angular momentum is ultimately responsible for everything we see around us.
"But wait," you object conveniently for my narrative, "You said that p is the linear momentum of a particle. How does something moving in a straight line have anything to do with rotation?"
It's true, the connection is not obvious-- objects moving in straight lines seem qualitatively different than objects that are spinning in place. But in the right circumstances, you can convert linear motion into rotational motion. If you throw a ball at a door, for example, and hit it in the right place, you can make that door swing open or closed. That interaction converts the linear motion of the ball, which flies through space in a straight line, into rotational motion of the door, which pivots about its hinges. This implies that even though it was moving in a linear fashion, the moving ball had some rotational character. That's what you calculate with today's equation.
"Yes, but that still doesn't look anything like an object spinning about some axis," you say. "The ball is moving through space, but a spinning wheel just sits there. They're completely different."
On a macroscopic level, they certainly look different, and it might not seem like this equation does you any good in describing a spinning wheel. But when you think about it, a spinning solid object is itself made up of mind-boggling numbers of individual atoms. At any given instant, each of those atoms is moving in a straight line through space, and thus has some (tiny) linear momentum. You can use that tiny linear momentum and the distance from the axis of rotation to calculate the individual atom's angular momentum, and then repeat the calculation for each of the other atoms. Add up all 1026 of those individual atomic angular momenta, and that's the angular momentum of the whole macroscopic object.
So, you can see, this is the most deceptive of all the equations we've looked at. Not only is it hiding scary vector business behind a pleasant facade, but it's vastly more powerful than it lets on. So, take a moment to celebrate the existence of angular momentum, say, by doing a little spinning dance, and we'll be back tomorrow with the next equation oft he season.
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One of the things that makes it hard to think about is that, unlike linear momentum, which is just computed relative to the velocity of some reference frame, angular momentum is also relative to the point you choose as the spatial origin of your coordinate system. If the linear motion of a particle is off-center such that the extrapolated straight line doesn't pass through the origin, that's what makes it have nonzero angular momentum. If the line does pass through the origin, then there is none (relative to that origin).
And this, I think, is the key to understanding the relationship between linear and rotational motion. In the swinging-door example, the door should be hinged on an axis passing through the origin.
An excuse to tell my two favorite dork jokes.
1)what do you get when you cross a mosquito and a mountain climber?
You can't, one's a vector and the other a scalar.
2) alright, what do you get when you cross an elephant and a tiger?
Elephant tiger sin theta
Both work better if you are with a bunch of drunk science/engineering undergrads...
And this, I think, is the key to understanding the relationship between linear and rotational motion. In the swinging-door example, the door should be hinged on an axis passing through the origin.
Yeah, I was planning to talk a little more about that when I get to torque (projected to be Saturday's entry). The swinging door thing is one of my favorite examples for torque, because most people can sympathize with the experience of walking toward one of those all-glass banks of doors and guessing wrong about which end has the hinge...
Perhaps it was stated earlier in this series, but I'm wondering what date is assumed for Newton's birthday, 12/25 or 1/4. Per Wikipedia, "During Newton's lifetime, two calendars were in use in Europe: the Julian or 'Old Style' in Britain and parts of northern Europe (Protestant) and eastern Europe, and the Gregorian or 'New Style', in use in Roman Catholic Europe and elsewhere. At Newton's birth, Gregorian dates were ten days ahead of Julian dates: thus Newton was born on Christmas Day, 25 December 1642 by the Julian calendar, but on 4 January 1643 by the Gregorian."
Perhaps it was stated earlier in this series, but I'm wondering what date is assumed for Newton's birthday, 12/25 or 1/4.
For the sake of the advent calendar joke, I'm taking his birthday to be Christmas Day. I'm not sure I'll have the stamina to make it through 24 equation posts, let alone 34.
Oh, surely there are scarier vector operators than the cross product.
Oh, surely there are scarier vector operators than the cross product.
We'll probably get to the Einstein equation from general relativity, which brings in some scary tensor stuff. But the cross product is the most innocuous-looking scary vector operation that we've seen to this point. I haven't decided yet whether to do Maxwell's Equations in differential or integral form, so I'm not sure whether the curl will turn up or not.
Curl is only scary because it's based on the cross product, wouldn't you say?
It took a long time for it to sink in, for me, that dot and cross products had to do with the "parallel part" and the "perpendicular part." I wish it had been taught with more drawing of triangles, less writing of letters.
I think I was in grad school, or anyway studying for the GRE, before I finally had any intuitive understanding of Div and Curl. (Meaning I could look at a picture of a vector field and tell you if its Div and Curl were zero or non zero, postivie or negative.) Again, I wish we'd practiced that skill earlier on, rather than concentrating on calculating them by manipulating symbols.
You might also mention that just like conservation of linear momentum was connected to the translational symmetry of the universe (i.e. the laws of physics are independent of location) the angular momentum is connected to the rotational symmetry. The laws of the universe are the same no matter which direction your facing.
I just wanted to take the time to thank you for this very cool series - The Advent Calendar of Physics. The articles are a great read and easy to grasp for a laymen such as myself. Your blog and others like it are the fanatastic little gems that make wading through the morass of vacuous internet drivel worth it.
The radius vector comes before the momentum vector in the cross product, so the angular momentum vector points straight down, not up.
Angular momentum can be made a vector in three-space (3 + 1, three spatial and one of time) because of the unique perpendicular vector to what AM really is: a planar 2-form, the plane in which the rotation occurs. For example, suppose we had four space dimensions and one of time, then the rotation in the x,y plane has no unique perpendicular. Instead, (not using exotic extra refinements like 4-vectors) define
Lik = Σ ri pk - rk pi
i,k = 1,2,3, ... n
Compare the exotic implications for rotation in n-space: a hyperplanet in spatial x,y,z,w space can be rotating at one rate within x,y and another rate within z,w: makes for strange days!
"Fine minds make fine distinctions."