Just a cool high school problem

[wrobel]

Thought some teachers might find this useful.

A wheel of radius $R$ rotates in a vertical plane about its fixed center. A speck of dirt detaches from a point on the wheel and, after a time interval $T$, impacts the wheel at the same height. Determine the angular velocity of the wheel. The acceleration due to gravity is $g$.
Find all possible answers.

 

[berkeman]

Find all possible answers.

Spoiler

I can see a series of solutions, but they all have the speck leaving the rim flying up vertically when the spot on the wheel is at 90 degrees from the vertical and the spot is moving up. The series of solutions are for one wheel rotation until re-impact, two rotations, and so on. Are there solutions other than from that starting position?

 

[wrobel]

Only two solutions: the point moves strictly vertically up and down, or it traces a parabola.

 

[Halc]

It seems that any angular velocity will do so long as it's high enough to involve centripetal acceleration of at least g. The problem needs perhaps another bit of information to narrow this down.

 

[DaveE]

Only two solutions: the point moves strictly vertically up and down, or it traces a parabola.

Yes, so there is no unique answer for the purely vertical motion of the particle. It will hit the wheel at the same height it left without regard to the velocity. I suggest removing this trivial solution.

Anyway, just to clarify. "at the same height" has nothing to do with the same location on the wheel, it's just relative to an external observer? You didn't mean at the same location on the wheel, right?