I Polar coordinates of a vector

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1. Sep 10, 2016

Precipitation

Note: All bold and underlined variables in this post are base vectors

I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The wheel rotates with uniform angular velocity dθ/dt = ω radians per second about an axis fixed in space.
At t = 0 the spoke is along the x axis, and the bead is at the origin. The book then says that the velocity of the bead at time t in polar coordinates is ur + uωtθ. Elaborating, the text says "at time t, the bead is at radius ut on the spoke."

What I don't understand is why u can be used in this calculation without any modification. If the bead is at radius ut at time t then the velocity would increase indefinitely and the spoke would have a position vector longer than the wheel it was attached to, which obviously doesn't make sense. Am I misunderstanding something about polar coordinates/vectors here or am I misunderstanding the example?

2. Sep 10, 2016

PeroK

Obviously, eventually the bead will reach the rim of the wheel. That equation is only valid until then.

3. Sep 11, 2016

Precipitation

That makes sense. I was conceptualising it as the bead reversing direction as the wheel completed successive revolutions. Thanks for the help.