How Do Polar Coordinates Explain a Bead's Velocity on a Rotating Wheel?

[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?

 

[PeroK]

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?

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

 

[Precipitation]

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

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