Stewart 16.5 #37 (a)
Let be a rigid body rotating about the -axis. The rotation can be described by the vector where is the angular speed of , that is, the tangential speed of any point in divided by the distance from the axis of rotation. Let be the position vector of .
By considering the angle in the figure, show that the velocity field of is given by .
I'll try to produce the figure mentioned in the problem if anybody wants it, but basically is the measure of the angle between and .
Now part of my problem is that I know jack about physics so I don't really know what a velocity field should represent. Angular speed? Tangential speed? I'm not totally clear on what the difference between these are anyway. Seriously, I know nothing about physics, I haven't taken a single class in it. But I guessed about this much:
Half the battle will be done if I show that speed at the point is but something here is clearly confused (bah-dum-ching) since I have no guarantee that .
The other half of the battle should be done by noting that the cross-product of of the two vectors will be normal to the plane formed by them thus pointing the vector in the appropriate direction. That's not exactly rigorous but I'm kind of okay with that at 1:30 AM.