Preview Activity 13.7.1.
We would like to understand and measure rotation of a vector field near a particular point. In order to investigate this concept, we will look at some two-dimensional vector fields and think about whether the vector field shown will rotate a small pinwheel or spinner placed at a particular location. The sort of spinner we imagine is illustrated in Figure 13.7.3. It consists of a central axis with a four-bladed paddle placed at one end of the axis. We imagine that the spinner is anchored at a point and the vector field, perhaps thought of as a fluid flow or wind velocity vector field, pushes against the blades of the spinner’s paddle. In this activity, we will be trying to assess how the the spinner will rotate around the black axel (as an axis of rotation).
To begin our investigation of the rotation of a spinner in a vector field, we will look at the vector field \(\vF\) in Figure 13.7.4. As you think about these questions, draw an “X” at each of the points about which you are asked and consider the vector field as being the pattern of a wind blowing across the plane.
(a)
Draw an X at the origin to act as your spinner. Draw a vector on the top right blade of your spinner that represents how the wind will push on that blade. Next, draw a vector on each of the other blades of your spinner that represents how the wind will push on that blade.
(b)
Use your vector representations from the previous part to describe if a small spinner placed at the origin would spin clockwise, counterclockwise, or not spin at all.
(c)
Now we would like to look at the rotational strength of \(\vF\) at the point \((1,1)\text{.}\) As before, draw a spinner at this point and draw vectors on each of the blades to represent how the wind will push on that blade. It is important at this stage to draw the relative lengths of the vectors on each blade to scale so you can see which blades have a larger force due to the wind.
(d)
Use your vector representations from the previous part to describe if a small spinner placed at at \((1,1)\) would spin clockwise, counterclockwise, or not spin at all. You should pay attention to which blades will have a stronger force (in comparison to the blade on the other side).
(e)
Would a small spinner placed at \((-2,1)\) spin clockwise, counterclockwise, or not spin at all? Draw a representation of a spinner if necessary to demonstrate your ideas.
(f)
Are there any points on the \(xy\)-plane where the spinner would not turn counterclockwise?
