$40$ points$)$ A set of students are visualizing flow

over a spinning baseball. Lacking an air tunnel, the

students do have a water tunnel in their laboratory

where they can inject multicolored streaklines to

visualize the flow. In order to visualize the flow,

they need to ensure similarity between the water

and the air. In the air, the baseball will move at a

velocity of $145$ km$/$h and spins at $300$ rpm $($note

that the spin rate can also be called the frequency,

f$,$ with dimensions of $\{1/$time$\}).$ The diameter of the

ball also matters here.

a$.$ Determine the functional relationship to establish dynamic similarity.

b$.$ What speed should the water be and what will the resulting rpm of the baseball in the water

be$?$

Properties of air: $=1\mathrm{.}204$ kg$/$m$3$ $=1\mathrm{.}825$ x $10-5$ kg$/$ms

Properties of water: $=1000$ kg$/$m$3$ $=1\mathrm{.}00$ x $10-3$ kg$/$ms