Curl of velocity vector in rotational motion

[brotherbobby]

TL;DR

In rotational motion, $\boxed{\vec v = \vec\omega\times\vec r}$, irrespective of whether the angular velocity $\vec\omega$ is constant or not (I hope I am correct in saying this). However, the curl of velocity vector, $\mathbf{\vec\nabla\times\vec v=2\vec\omega}$ comes out only to be true if the angular velocity of the rotating body is $\vec\omega$ constant. Is this true? I show the details below.

Attempt : I draw the diagram of the problem to the right. A circular plate of radius $a$ rotates with an angular velocity $\vec\omega = \omega_0\hat k$, assumed constant for the moment. A particle P at rest with the plate lies on the rim along the $x$ axis, as shown. Its velocity is $\vec v= v\hat j$ and its position vector $\vec r = a\hat i$. Using the right hand rule, these vectors make sense : $\vec v = \vec\omega_0\times\vec r\Rightarrow v=\omega_0 a$, remembering that $\hat j = \hat k\times\hat i$. Calculating the curl of this vector using index notation, $\small{\vec\nabla\times\vec v = \vec\nabla\times(\vec\omega_0\times\vec r)=\epsilon_{ijk}\partial_j(\vec\omega_0\times\vec r)_k\hat e_i = \epsilon_{kij}\epsilon_{klm}\partial_j\,\omega_{0_l}\,x_m \hat e_i=(\delta_{il}\delta_{jm} -\delta_{im}\delta_{jl})\partial_j\,\omega_{0_l}\,x_m \hat e_i}$, which reduces to $\small={\partial_j\,\omega_{0_i}\,x_j \hat e_i-\partial_j\,\omega_{0_j}\,x_i \hat e_i=3\vec\omega_0-\vec\omega_0=2\omega_0}\Rightarrow \boxed{\vec\nabla\times\vec v=2\vec\omega_0}$, where I have used the fact that $\vec\omega_0$ is uniform and $\partial_i x_j = \delta_{ij}$. This can also be shown using the method of determinants for the cross product.

Question : Clearly, from my derivation, the angular velocity $\vec\omega_0$ is constant. Does it have to be? In which case, I am mistaken in what I have shown. Can it not be that the curl of the velocity vector at a given time $t$ equals twice the angular velocity at that time : $\vec\nabla\times\vec v(t)=2\vec\omega(t)$?

 

[A.T.]

However, the curl of velocity vector, $\mathbf{\vec\nabla\times\vec v=2\vec\omega}$ comes out only to be true if the angular velocity of the rotating body is $\vec\omega$ constant. Is this true?

Constant with respect to what? Time or position? If $\omega$ varies with respect to r, then curl is different than when it doesn't. If $\omega$ varies with respect to t, then so will the curl.

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