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Angular Momentum: Mass Rotating About a Post

  1. Jul 16, 2009 #1
    1. The problem statement, all variables and given/known data

    This question is from Kleppner and Kolenkow's mechanics book, problem number 6.13:

    Mass m is attached to a post of radius R by a string. Initially it is distance r from the center of the post and is moving tangentially with speed v0. In case (a) the string passes through a hole in the center of the post at the top. The string is gradually shortened by drawing it through the hole. In case (b) the string wraps around the outside of the post.

    What quantities are conserved in each case? Find the final speed of the mass when it hits the post for each case.

    2. Relevant equations



    3. The attempt at a solution

    Okay, in part (a), I believe I have the answer, but I would like to make sure. Since the rope is being pulled through a hole at the center of the post, the tension force is central (radially inward). Therefore the torque about the center of the post must be zero; from there, we may conclude that angular momentum (mvr) is conserved. Thus, v0r = vfR and the problem is solved.

    Part (b), however, is different. Since the string is wrapping around the loop, the tension is not radial; thus, there is a torque on the mass about the center of the post, and angular momentum is not conserved. Clearly neither linear momentum nor energy are conserved. From here, I am not sure where to go. I could use the work-kinetic energy theorem, but this would involve a line integral about a path that I do not know...

    Any help on this problem would be greatly appreciated. Thanks.
     
  2. jcsd
  3. Jul 16, 2009 #2

    tiny-tim

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    Hi Sam_Goldberg! :smile:

    (a) is ok (though you haven't mentioned why energy is not conserved)

    (b): again, is energy conserved?

    Hint: what is the work done ? :wink:
     
  4. Jul 16, 2009 #3
    Well, in part (a), the force is always radially inward and the ball moves in such a way that it does have a radially inward component of motion, so F dot dr is nonzero and work is done on it. That's why energy is not conserved. In part (b), however, I get the feeling you're trying to tell me that the motion is always perpendicular to the rope. I see how this is initially true (when you stretch out the rope and give the ball its initial boosh to give it the speed v0). Yet, I do not see how this can continue...

    Wait a minute: if the mass had a component of motion along the rope, then I bet the rope would collapse. In part (a), an external person was coming in and shortening the length of the rope, thus allowing the mass to have a component of motion along the rope. But in part (b), no one is doing this. Does this reasoning show that the mass in part (b) cannot have an F dot dr, and thus no work is done? Then the velocity would never change, of course. Is this correct?
     
  5. Jul 17, 2009 #4

    tiny-tim

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    :biggrin: Woohoo! :biggrin:

    Yes, instantaneously, the velocity is always perpendicular to the rope. :smile:
     
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