The rotation speed for a density that varies with 1/r doesn't seem too natural, however. It apears to climb, approximately with 1/sqrt(r), so gets larger toward the center. That would be because the density at the center is 1/r, so approaches infinity there. A graph of the density would be similar to a hyperbola, and climb to infinity at the center, so I will just top it off there.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."