) A vortex that rotates at constant angular velocity w about the z-axis has velocity vector field v⃗ =w(−yi⃗ +xj⃗ ). (a) Sketch, on a separate sheet of paper, the vector field with w=1 and the vector field with w=−1. Then determine the speed ∥v⃗ ∥ of the vortex as a function of the distance from its center, r. speed

Explanation:

check below for answer and explanations to your question

The graphical relationships and the relationship between the linear and angular variables allows finding the results for the movement of the vortex are:

A vortex is a circular motion where the angular speed of the particles is constant.

They indicate the function of the linear velocity of the vortex is:    

Where v is the linear speed, w the angular velocity and (y, x) the position of the particle.

a) In the attachments we can see the linear velocity for the cases:

b)   The linear and rotational variables are related.      

               v = w r

We can find the distance with the Pythagorean theorem.  

             r =

Let's substitute.

In conclusion using the graphical relationships and the relationship between the linear and angular variables we can find the results for the movement of the vortex are:

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