# Rotation : angular velocity from body frame

1. Apr 21, 2004

### venkat

consider a body rotating with a constant angular velocity 'w' along the z axis of the space reference frame.the centre of mass of the body is at the origin of the space system of coordinates.

now since 'w' is a vector, it must transorm according to the rotation matrix
R (in this case as given below), when we pass from the space frame to the body frame (rigidly fixed on the body and with its z' axis along the z axis of the space frame).

R = | cos(wt) Sin(wt) 0 |
| -Sin(wt) Cos(wt) 0 |
| 0 0 1 |

here i have used (wt) in the place of the angular seperation (theta) between the two reference frames, as the angular seperation increases linearly with time.

pardon me for using the symbol '|' for denoting the matrix.i have used this symbol because i couldn't find anything better on the keyboard.
even though it looks like a determinant, i mean a matrix.

when we use this matrix for transforming from the space frame to the body frame, we get the angular velocity as 'w' in the body frame also !
but,in the body frame, shouldn't the angular velocity be zero ?
(since the body frame is rotating with the body)

what is wrong? is the rotation matrix wrong, or can't the angular velocity be transformed according to the rotation matrix when we pass from space frame to body frame?

2. Apr 21, 2004

### venkat

sorry about the notation for the rotation matrix

it should be

R = | cos(wt) Sin(wt) 0 |
| -Sin(wt) Cos(wt) 0 |
| 0 0 1 |

3. Apr 21, 2004

### venkat

WHAT I MEAN IS THIS:

R11= Cos(wt)
R12= Sin(wt)
R13= 0
R21= -Sin(wt)
R22= Cos(wt)
R23= 0
R31 = 0
R32 = 0
R33 = 1

4. Apr 21, 2004

### Integral

Staff Emeritus

5. Apr 21, 2004

### HallsofIvy

Staff Emeritus
"when we use this matrix for transforming from the space frame to the body frame, we get the angular velocity as 'w' in the body frame also !
but,in the body frame, shouldn't the angular velocity be zero ?
(since the body frame is rotating with the body)"

The angular velocity of what? The matrix for "transforming from the space frame to the body frame" is the inverse of the rotation matrix itself- the same rotation matrix but with negative angle. The angular velocity of "space" relative to the body is -w.

6. Apr 22, 2004

### venkat

you said :

the angular velocity of what?

i mean the angular velocity of the body.

also, shouldn't the angular velocity of the body be zero in the body frame?

7. Apr 23, 2004

### venkat

since the transformation matrix R is orthogonal, you get the angular velocity as 'w' in the body frame also, since R' = $$R^-1$$

8. Apr 23, 2004

### venkat

since the transformation matrix R is orthogonal, you get the angular velocity as 'w' in the body frame also, since R' = R inverse

9. Apr 23, 2004

### venkat

can somebody help me? i have no one to discuss things with as i am studying physics on my own...that's why i put up the question here.

10. Apr 25, 2004

### Coelum

Late replay because I joiuned Physics Forum a few days ago.

Why should "W" change? It is the rotation vector computed in the fixed frame. If you want the rotation vector in the rotating frame, just compute it in the rotating frame and you'll get 0 as expected.

11. Apr 26, 2004

### venkat

i don't get 'w' as expected..try it!

12. Apr 26, 2004

### turin

venkat,
ω is not a true vector, it is a pseudo vector. You can see problems arise in a very formal way, as you have indeed shown. Or, you can endeavor a more heuristic analysis, and realize that, what you are calling ω does not really exist, but, if it did, it would in fact be an EIGENVECTOR of your proposed transformation matrix. The eigenvalue is unity (as can clearly be seen), so there is no way for this matrix to transform this vector into anything else.

The problem is that, ω was constructed artificially in the first place, and then the issue is compounded by the fact that its direction is used to determine the proposed matrix. Off the top of my head, I would suggest treating the rotating body as a composite of points each with its own individual position and velocity vectors. These could all be resolved into an angular velocity (pseudo-)vector, but then you inherently lose some information. Again, this is a heuristic analysis, but it seems reasonable, since, by beginning with two vectors, x and v, and then declaring ω as the total character of the system, it seems that two vectors have been reduced to one. Further more, these two vectors, x and v, seemed to be independent of each other, that is, you are allowed to specify (kinematically) any v you like at any x. Of course, this specification will dictate strict requirements on the other points in the body (if it is rigid and fixed at the origin), but the fact still remains that a declaration of ω is a reduction of two independent vectors, v and x, into only one. Something is certainly missing.