Damn you mathematicians - Page 3

Fels krone · #21 03-01-2007, 01:05 PM

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'cause s doesn't have units length*angle. Or at least that's my guess.

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Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

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I still don't understand what is your problem with the given expression. That must make me pretty dumb, since I "do" rotational kinematics for a living. (I design attitude control systems for spacecraft.)

FWIW, where possible, I prefer to express 3-dimensional rotational kinematics in terms of the rate of change of a quaternion: (see equation 9, p. 14)

Quaternion dynamics

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Attitude Control Systems, heh heh

Im not sure where the confusion is, radians are unitless. If you dont like that, go back to geometry and convince yourself of why they have no units. I learned why a long time ago, but I dont worry about it every time I have to work a problem with radians.

Magic_Man · #22 03-01-2007, 01:08 PM

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The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

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To be fair, it's only "correct" for one system of LINEAR measurement also. S has to be the same units as R. Doesn't seem any more arbitrary than saying that theta has to be in rads.

Borodog · #23 03-01-2007, 01:12 PM

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'cause s doesn't have units length*angle. Or at least that's my guess.

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Bingo. The expression is dimensionally incorrect. I [censored] HATE THAT [censored].

Me for 1/2 the semester: UNITS UNITS UNITS UNITS!

Me 1/2 way through the semester: Uh, except . . .

[/ QUOTE ]

I still don't understand what is your problem with the given expression. That must make me pretty dumb, since I "do" rotational kinematics for a living. (I design attitude control systems for spacecraft.)

FWIW, where possible, I prefer to express 3-dimensional rotational kinematics in terms of the rate of change of a quaternion: (see equation 9, p. 14)

Quaternion dynamics

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Attitude Control Systems, heh heh

Im not sure where the confusion is, radians are unitless. If you dont like that, go back to geometry and convince yourself of why they have no units. I learned why a long time ago, but I dont worry about it every time I have to work a problem with radians.

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A radian is emphatically NOT unitless; it is physically dimensionless. You can tell that it is not unitless because it has a unit, i.e. 1 radian is distinct from 2 radians.

bluesbassman · #24 03-01-2007, 01:13 PM

Okay, now I understand what you are saying. I would simply teach that, by definition, all variables which represent angles in mathematical expressions are radians.

I agree though that you need to be picky about units. If you want to really be a jerk (like me), check your bill for units next time you eat out at a restaurant. Often you will find there is a number shown which apparently represents what you owe for the meal, but includes no monetary units or dollar symbol. It's then great fun to sincerely ask the server whether that number represents amount owed in dollars (or whatever). If you are dining with a wife/girlfriend, she will get REALLY mad at you. Yes, I have actually done this. [img]/images/graemlins/blush.gif[/img]

(But at least I tipped the server extremely generously for putting up with me.)

Borodog · #25 03-01-2007, 01:22 PM

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The very fact that the expression s = r*theta is only "correct" for one system of angular measurement tells you that there is monkey business going on.

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To be fair, it's only "correct" for one system of LINEAR measurement also. S has to be the same units as R. Doesn't seem any more arbitrary than saying that theta has to be in rads.

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In fact, no. For example, 1mi = 1609m.

Duke · #26 03-01-2007, 01:30 PM

Boro,

So you're really annoyed by unit-less angles in principle, or you're just annoyed that it makes the teaching job harder? I'm actually having a difficult seeing it through your eyes, since I guess I internalized the concepts much differently.

I'm fairly certain that introducing a magic constant of 1/rad would be far more confusing to everyone than just realizing that a radian is just a pure number.

Borodog · #27 03-01-2007, 01:42 PM

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Boro,

So you're really annoyed by unit-less angles in principle,

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Angles are emphatically NOT UNITLESS; rather they are physically dimensionless.

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or you're just annoyed that it makes the teaching job harder? I'm actually having a difficult seeing it through your eyes, since I guess I internalized the concepts much differently.

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a) It undermines everything that I have taught the students about units and dimensional analysis to have units that magically disappear when no other units have this property, and

b) I find it extremely aesthetically displeasing.

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I'm fairly certain that introducing a magic constant of 1/rad would be far more confusing to everyone than just realizing that a radian is just a pure number.

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I doubt it. We introduce "magic constants" all the time in physics, and we always construct them so that the units work properly. For example, G = 6.67x10^-11 Nm^2/kg^2.

Duke · #28 03-01-2007, 01:52 PM

Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

If you tell someone that you want 3/5 of a pizza, and the Pizza has 8 slices, you'll be getting 4.8 slices. Certainly not 4.8 FRAC*slices*1/FRAC.

Perhaps the equivalence I see between these is illegitimate.

EDIT: Fixed middle paragraph so that the right guy is getting the pizza.

gumpzilla · #29 03-01-2007, 02:02 PM

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Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel. I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

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This is the definition of a radian - the angle such that the arc length subtended by that angle on a circle is equal to the radius.

Boro - I see what you're saying. But I think if you just explicitly mention that radians are essentially just 2*pi times a fraction of a circle, it becomes pretty clear that they are dimensionless, AND that this is what you'd want for expressing lengths.

Borodog · #30 03-01-2007, 02:07 PM

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Well, if you want to pull a fast one on the students, you'll just write it as theta = s/r and they'll see that the units cancel.

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Fine. Where then do the magical radians come from? It's simply inconsistent and hence annoying.

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I think that textbooks tend to do that anyhow. I vaguely recall it being first introduced like that in HRK.

If you tell someone that you want 3/5 of a pizza, and the Pizza has 8 slices, you'll be getting 4.8 slices. Certainly not 4.8 FRAC*slices*1/FRAC.

Perhaps the equivalence I see between these is illegitimate.

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I don't see any equivalence at all.