String and Pillar

A string of length \(L_o\) is tied to a circle of radius \(R\) and a very tiny ball. The ball is given an initial angular velocity \(\omega_o\) perpendicular to the string. The string is initially perpendicular to the radius from the center of the circle to the point where the string is tied to it. The string wraps around the circle and there is no friction or gravity. How long does it take, in seconds, for the ball to touch the circle. Input your answer to the nearest hundredth.

Details and Assumptions:

  • The string (red in picture) has effectively 0 width and 0 mass.
  • The values of \(R\) and \(L_o\) are such that that ball hits the circle.
  • The circle (black) doesn't move and has radius \(R=\frac 12 L_o\).
  • The ball (blue) has a negligible radius but still has mass.
  • \(\omega_o=\) 0.25 rad/s
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