The angular velocity vector of a rigid object rotating about the z-axis is given by
ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point.
a.) Assuming that ω is constant, evaluate v and ∇ X v in cylindrical coordinates.
b.) Evaluate v in spherical coordinates.
c.) Evaluate the curl of v in spherical coordinates and show that the resulting expression is equivalent to that given for ∇ X v in part a.
The expressions for the curl in cylindrical and spherical coordinates. Since I don't know how to put the determinant here ill just leave them out.
x = r sinθ cosΦ
y = r sinθ sinΦ
z = r cos θ
The Attempt at a Solution
So I worked out part a correctly (I think) which is in the attachment, but I'm stuck on part b.
b.) So for this part I have to convert ω to spherical coordinates. Since ω only lies along the z-axis, that means that Φ and θ are equal to zero, so
ω = ω r-hat
and the position vector in spherical polars is
R = r r-hat
so that means that when I cross ω and R I get zero, I don't know what I'm missing.
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