Is torque normal to the plane of rotation? For god's sake! Why?!

Factual Questions
1

I’m gonna die if I can’t understand this. It is literally killing me. Cough, cough, gasp!

Right hand rule: We have a lever working in the plane. One end is fixed, the other is free to move so that if I apply a force to the free end, perpendicular to the lever, it will spin around the fixed point. Okay. I point the fingers of my right hand in the direction of the lever, going from the fixed end to the movable end, and curl my fingers in the direction of rotation and I will see that my right thumb points in the direction of the torque vector. E.g. if I’m turning a bolt counter-clockwise, I point my fingers along the wrench from the bolt to the handle and curl my fingers counter clockwise and, if the bolt is on a table top, my thumb (and the torque vector) point straight up.

Do I got that right?

I understand that the vector is the cross product of force and length of the lever arm (or something like that) which is also the dot product of force, lever arm, and the sine of the angle through which the lever is rotated (or something like that).

What I absolutely do not get, perhaps because I never actually studied trig in any formal setting (I only managed to survive what it was assumed I had learned when we used trig stuff in calculus & whatnot) and I had totally cratered linear algebra (yet somehow managed to pass the course :rolleyes: ), is why the torque vector would end up normal to the plane of rotation. Is there a simple non-mathematical explanation?

Going a step further, this vector seems to be really real. What?! Suppose I have a spinning wheel, I can apply the right hand rule to get the direction of the torque. Knowing that, if I rotate the spinning wheel perpendicular to its plane of spin, I can apply the right hand rule using the torque vector as my lever arm (I guess) to predict which way the whole shebang will want to move as a result of gyroscopic what’sits. What? Why is this mathematical construct, a cross product, moving real stuff in my physical world?!

If I ever need to remind myself that I am not a solipsist, I play with my gyroscope—'cos there ain’t no way my mind could make that shit up.

Can anyone make me understand? This has been driving me nuts for about six years.

2

Let me take a shot at it: (and apologies if this is annoyingly simple)
Let’s just think about a rotation vector, because it’s a little simpler than torque, but the same idea.
What’s real is the object that is rotating, and you also could say that the rotation itself is real. The vector isn’t real in the same way; it’s just an easy and useful way to describe the rotation. Why a vector? Because what we want to describe is the plane that the object is rotating in, and how fast.
To describe a plane, we could give three points (which is enough to define a plane) or a line and a point, or whatever, but we could also give a vector perpendicular to the plane, which also defines a plane. The advantage of this is that since the length of the vector doesn’t matter for defining the plane, we can use the length of the vector to describe how fast the rotation is, and in one simple term, we’ve described both the plane of rotation and the speed. No deep physical reason, it’s just easy.
The only issue is which way the vector points, because up or down pointing arrows define the same plane. So mathematicians and physicists just abritrarily decided to use the right hand rule (kind of like arbitrarity deciding to drive on the right in the U.S., or writing numbers from right to left).

So the rotation or torque vector doesn’t really exist, it’s just a very useful way to describe rotation. You can think of it as showing the axis of rotation, if you want some physical interpretation, but that’s not really necessary.

We use the torque and rotation vectors because they are useful, as in a calculating a gyroscope’s movement. Remember, it’s not some quasi-real torque vector moving the gyroscope, it’s a real force (gravity, or your hands pushing on the axle of the bike wheel or whatever). It’s just that it gets very complicated figuring out the force of gravity at every point, and what’s cancelled by the forces holding the wheel together and the acceleration at every point and what the velocity is at every moment, etc. It turns out that using some simple rules with torque and velocity vectors gives you the right answer with a lot less work, so we use the vectors.
The right hand rule is arbitrary, but as long as we use the same rule when going from the vector back to rotation as we did going from rotation to vector, it works out and gives the right answer. Which is all physicists and engineers really care about, after all.

3

I don’t think I understand. Shouldn’t torque be anticlockwise against clockwise rotation? You seem to be suggesting that it is perpendicular to the plane of rotation, or maybe I just don’t follow your post.

4

Quercus hit it on the head, I think. I’ll just add that the fact that we can even think of the angular momentum as a vector is an accident of the number of spatial dimensions we live in. The number of independent directions of rotation you can have is, in general, d*(d-1)/2, where d is the number of dimensions. So in two dimensions, you only have one possible rotation (in, say, the xy plane); in three dimensions, you have three (in the xy, yz, xz planes); in four dimensions, you have six (in the xy, yz, xz, wx, wy, wz planes); and so forth. Three dimensions happens to be the only number of dimensions for which d*(d-1)/2 = d, so we can define a convenient correspondence between planes of rotation (defined by two vectors lying in the plane) and vectors, which is — you guessed it — the cross product.

Sorry if I’m getting to convoluted here. I had to restrain myself from talking about p-forms, the Hodge star operator, and representations of SO(N) here, so I think you got off pretty easy. :slight_smile:

5

Nope, you got it. If you lay a plate flat on a table and rotate it to counter-clockwise, the torque vector will point straight up out of the table. Since the vector is perpendicular to the table, since it points up (or down), and since it has a certain length, it tells us all about the rotation of the plate. So you give an engineer a torque vector, and he can tell you in what plane the rotation is happening, in what direction, and (presumably) the length of the lever arm and how far it is rotated.

I had always thought that this vector was somehow a physical thing, but it’s not. It is a “linguistic” convention, for lack of a better word. But since the “language” we’re using is math, it is an internally consistent, logical framework that we can use to draw conclusions & predictions. Hence, I can use the “language” to predict the behavior of my gyroscope.

I had thought that the torque vector was a new lever arm, so to speak, and by moving that lever arm we got the behavior of the gyroscope. But that’s not the case; the behavior is the product of other physical stuff and the torque vector is a useful descriptive tool for working through what happens.

Brilliant. I can not begin to describe how much this has vexed me.

6

If it helps any, torque isn’t strictly speaking a vector, it’s a pseudovector. Whenever you cross two true vectors or two pseudovectors, you get a pseudovector, but when you cross a pseudovector and a true vector, you get a true vector. Pseudovectors can be recognized by the fact that they always need a right-hand rule to define their direction.

As MikeS says, it’s only because we have three dimensions that we can get away with the notion of pseudovectors. In any other number of dimensions, we need to use more complicated objects like antisymmetric tensors and wedge products, instead of pseudovectors and cross products. Never mind just what those things are; just take my word that they’re more complicated, and be glad that we have three dimensions.

7

[QUOTE=js_africanus]
Going a step further, this vector seems to be really real. What?! Suppose I have a spinning wheel, I can apply the right hand rule to get the direction of the torque. Knowing that, if I rotate the spinning wheel perpendicular to its plane of spin, I can apply the right hand rule using the torque vector as my lever arm (I guess) to predict which way the whole shebang will want to move as a result of gyroscopic what’sits.

[QUOTE]

This bit here makes me think that js_africanus is confusing the torque vector and the precession effect. And while quercus and MikeS have done an excellent job explaining the former, the latter remains unaddressed.

8

D’Oh!

Preview, preview, preview, preview…

9

Ok, I understand. So you’re not talking about the actual torque itself, but the vector used to describe the torque, and the vector doesn’t actually point in the direction the torque works in, but it still does a perfectly good job of describing it!

10

What I think the OP was referring to is the fact that precession can be explained by assuming that angular momentum and torque are “real” vectors obeying the equation tau = I * d omega /dt. This can be explained without using pseudovectors; the only source I know of where this is done is The Cartoon Guide to Physics by Larry Gonick. He explains things the “conventional” way but then also does a qualititative explanation without using the vectors. (I highly recommend this book in general, BTW.)

While this method is conceptually useful, I doubt that you could get any useful quantitative information (e.g. precession frequency, nutation) out of it. Like the OP said, the torque vector may be a mathematical fiction, but it’s a very very convenient one.

11

I used to have a bookmarked video of a physics prof. explaining it using a bicycle wheel dangling from the ceiling. No vectors nor pseudovectors. I’ve never been able to relocate it after the hard drive cratered.

Took another try…no luck. But you may enjoy these stats related lectures (the Benford’s Law one is fascinating):

www.dartmouth.edu/~chance/ChanceLecture/AudioVideo.html#Videos00

While I’m at it:

www.dartmouth.edu/~chance/chance_news/news.html

Yeah, but just think of how much fun tether “ball” would be in 4-space.

12

No, the simple answer is mathematical. The best one can do without math is to say that the torque “vector” is a fiction that gives the right answers.

13

Not in any perjorative sense. The consumer doesn’t set up and solve a Lagrangian function to maximize her utility; utility per se being a mathematical function of her preferences for goods considered in pairs. I’m on-board with mathematical fictions; however, sometimes it is hard to tell where the math that has been closely attached to “reality” (e.g. I have two apples and I give you one), splits off to become an accurate and binding logical tool that doesn’t necessarily correspond to something “real”. I didn’t realize that I had passed that point.

If I’ve been put correctly back on track, the torque pseudovector is real in the sense that it is a meaningful consequence of living in a logical (3-space) universe. While some conventions may be arbitrary, e.g. the right hand rule, physical things (often) can be mapped to mathematical things and those things have logical consequences that predict what physical thing can or will do. The torque pseudovector is one of those logical consequences; but, it is not a new magical lever-arm that has appeared from nowhere.

But what is the simple mathematical answer?

14

The “pseudovector” is a gimmick. It’s called that because it transforms like a vector under rotations of space, but opposite to a vector under reflections. It’s a purely physics term.

As others have alluded to, torque is really a 2-form in three-dimensional space. There is a map from p-forms in n-space to (n-p)-forms in n-space called the “Hodge Star” (also mentioned earlier) which gives a form which is perpendicular to the original and has considerations for length and orientation. The torque “vector” is the star applied to the torque 2-form, giving a 1-form, which is roughly equivalent to a vector for our purposes.

The torque 2-form itself is visualized as a flat chunk of the plane in which the rotation is taking place. The area of the chunk is proportional to the magnitude of the torque, but there is no information as to its shape. It’s just “this much of that plane with this orientation”. The star gives a perpendicular, which is why the torque vector is perpendicular to the plane of rotation.

To sum up, the simple answer (given the mathematics of multilinear algebra) is this: Torque is most conveniently expressed as a 2-form, which physicists long ago conflated with a perpendicular vector.

15

why think of it as normal to plain of rotation, think of it as lying on the axis of rotation.

it could not possibly lie in the plane of rotation cuz then how would you decide which way to point it ?

the way it is you only have two possibilities, clockwise or counterclockwise - makes perfect sense.

after all its all about accountability, it does not have to be something you can visualize.

16

That’s part of the definition of a 2-form: its orientation. It doesn’t really “point”, though, which is only applicable to the orientation of a 1-form (“covector”).

Unless, of course, by “it” you meant “the torque vector”. You see, I really can’t dereference pronouns through magic. I’ve done the bast I could with context, but you really need to quote things or put other explicit nouns in your text to make yourself clear. If you did mean the torque vector, go back and read my post again before replying to it with such a non sequitur.

17

How does that fit into the discussion in Comments on Staff Reports about whether there are three dimensions or not?

18

You write as though the torque “vector” is some sort of sleaze, yet here you seem to state that the torque 2-form doesn’t supply enough information and the star is utilized to correct that short-fall.

19

Torque (as it’s being used here) part of Newtonian physics, in which there are three spatial dimensions. There’s an extension of the concept to SR but it is, as stated, significantly more complicated.

20

The “torque vector” has three real components, just like the torque 2-form does. They contain the same information. The whole picture written out in the language of forms is simple and elegant, but the concept of torque arose (and is now taught) before the language of forms was available.

A force (F[sub]1[/sub],F[sub]2[/sub],F[sub]3[/sub]) applied at a displacement (r[sub]1[/sub],r[sub]2[/sub],r[sub]3[/sub]) from the pivot induces a torque r x F on a body. The modern geometrical viewpoint shows that A x B is really *(A^B), where * is the Hodge operator and ^ is the antisymmetric product.

r x F = *(r^F) =
*((r[sub]1[/sub]e[sup]1[/sup] + r[sub]2[/sub]e[sup]2[/sup] + r[sub]3[/sub]e[sup]3[/sup])^(F[sub]1[/sub]e[sup]1[/sup] + F[sub]2[/sub]e[sup]2[/sup] + F[sub]3[/sub]e[sup]3[/sup])) =
*((r[sub]1[/sub]F[sub]2[/sub]-r[sub]2[/sub]F[sub]1[/sub])e[sup]1[/sup]^e[sup]2[/sup] + (r[sub]1[/sub]F[sub]3[/sub]-r[sub]3[/sub]F[sub]1[/sub])e[sup]1[/sup]^e[sup]3[/sup] + (r[sub]2[/sub]F[sub]3[/sub]-r[sub]3[/sub]F[sub]2[/sub])e[sup]2[/sup]^e[sup]3[/sup]) =
(r[sub]2[/sub]F[sub]3[/sub]-r[sub]3[/sub]F[sub]2[/sub])e[sup]1[/sup] +(r[sub]3[/sub]F[sub]1[/sub]-r[sub]1[/sub]F[sub]3[/sub])e[sup]2[/sup] +(r[sub]1[/sub]F[sub]2[/sub]-r[sub]2[/sub]F[sub]1[/sub])e[sup]3[/sup]

where {e[sup]i[/sup]} is an orthonormal basis for R[sup]3[/sup].

Now this does put the torque into the form of a vector, but the Hodge operator has mangled the information about how the quantities should transform under spatial reflections.

Imagine reflecting space through the origin. Call this operation S.

S(e[sup]1[/sup]) = -e[sup]1[/sup]
S(e[sup]2[/sup]) = -e[sup]2[/sup]
S(e[sup]3[/sup]) = -e[sup]3[/sup]
S(e[sup]1[/sup] x e[sup]2[/sup]) = S(e[sup]3[/sup]) = -e[sup]3[/sup]
S(e[sup]1[/sup]) x S(e[sup]2[/sup]) = -e[sup]1[/sup] x -e[sup]2[/sup] = e[sup]3[/sup]

See? There’s a sign flip in there. This shows that the cross product of two vectors behaves unlike a vector under reflections. Physicists called it a “pseudovector”, but in the language of forms it behaves exactly as one would expect, since reflecting a 2-form flips two signs in each term, which cancels out.

Converting to simple vector notation loses the distinction between vectors and “pseudovectors”, while with forms it comes standard.