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I Polar coordinates of a vector

  1. Sep 10, 2016 #1
    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. jcsd
  3. Sep 10, 2016 #2

    PeroK

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    Obviously, eventually the bead will reach the rim of the wheel. That equation is only valid until then.
     
  4. Sep 11, 2016 #3
    That makes sense. I was conceptualising it as the bead reversing direction as the wheel completed successive revolutions. Thanks for the help.
     
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