7.1: Linear and Angular Velocity
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We related the linear and angular velocities of a rotating object in two dimensions in Section 5.1. There, we also already stated the relation between the linear velocity vector and rotation vector in three dimensions (Equation 5.1.5):
v=ω×r
It is not hard to see that this expression indeed simplifies to the scalar relationship v=ωr for rotations in a plane, with the right sign for the linear velocity. That’s hardly a proof though, so let’s put this on some more solid footing. Suppose r makes an angle ϕ with ω. Suppose also that it changes by dr in a time interval dt, then if we have pure rotation, dr is perpendicular to both r and ω, and its magnitude is given by |dr|=ωrsinϕdt=|ω×r|dt, where ω and r are the lengths of their respective vectors. Finally, as seen from the top (i.e., looking down the vector ω), the rotation should be counter-clockwise (by definition of the direction of ω), which corresponds with the direction of ω×r. We thus find that both the magnitude and direction of v=dr/dt indeed equal ω×r, and Equation 5.1.5 holds.