curl of linear velocity...?

Question

if a rigid body is rotated about a fixed axis , prove that the curl of the linear velocity 'v' of a point on the body is equal to twice the angular velocity...

Answer 1

It pretty much falls out of the definition. If v is a differentiable vector function such that
v=v1i + v2j + v3k (where i, j, k) are the basis of hte coordinate system) then (in a right-handed system)
curl v = (δv3/δy - δv2/δz)i + (δv1/δz - δv3/δx)j + (δv2/δx - δv1/δy)k
where I've used δ/δx to represent 'partial derivative with respect to x' and so on. Now the velocity field can be represented by
v=wk X r where r is the position vector of the point and the rotation is along the z-axis. Then just plug 'n chug to get the desired result.

Doug