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Pendulum/Circular Motion Problem

  1. Apr 12, 2014 #1
    1. The problem statement, all variables and given/known data

    A small ball of mass m is attached to a very light string of length L that is tied to a peg at
    point P. What is the magnitude of the horizontal velocity that must be applied to the ball
    so that it swings up and lands on the peg? Your answer can only contain the given
    information and any appropriate physical constants (such as g).
    If you have the acceleration of gravity g in your answer do not
    substitute numbers for it. You may introduce other variables to
    help you reach a solution but your final solution can only
    contain the given variables and g.
    (SEE ATTACHED)
    2. Relevant equations

    Conversation of Mechanical Energy?

    3. The attempt at a solution

    My initial thought was to set ΔK=ΔP since gravity is a conservative force and the only displacement is vertically, the length of the string. That gave me an initial velocity of √2gL
     

    Attached Files:

  2. jcsd
  3. Apr 12, 2014 #2
    Your solution only gets the ball horizontal with the peg. You need to give it more speed to cause it to land on top of the peg.
     
  4. Apr 13, 2014 #3
    Exactly. Thanks for the reply. I found an initial velocity to make a complete revolution also, which is not what I'm looking for. Any ideas anyone?
     
  5. Apr 13, 2014 #4
    And what velocity is that?
     
  6. Apr 13, 2014 #5

    haruspex

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    Clearly the answer lies between those two extremes.
    You need the string to go slack at some point above the horizontal, but before the vertical.
    Suppose it goes slack at angle θ above the horizontal. What initial velocity would lead to that (as a function of θ)? What will the subsequent trajectory be?
     
  7. Apr 13, 2014 #6
    √(2gL) ≤v_0≤ 2√(gL)
     
  8. Apr 13, 2014 #7
    Could I relate that to the tension in the string? Or centripetal acceleration?
     
  9. Apr 13, 2014 #8
    How did you come up with that?
     
  10. Apr 13, 2014 #9
    Yes, when the string goes slack the tension becomes________.
     
  11. Apr 13, 2014 #10

    Right, I'm just trying to find a way to relate them mathematically without introducing unnecessary variables because the solution can only contain the given variables.
     
  12. Apr 13, 2014 #11

    haruspex

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    It is often appropriate to introduce extra variables, then find enough equations to eliminate them. I would certainly recommend introducing one for the angle at which the string goes slack.
     
  13. Apr 13, 2014 #12
    I'm needing some direction. I realize that at some point after the horizontal the string must go slack; at this point it seems to be a projectile motion problem with a displacement in the x direction (L). Any thoughts?
     
  14. Apr 13, 2014 #13
    How would I do that?
     
  15. Apr 13, 2014 #14
    Vo=√(2gL(1-cosΘ))

    How am I coming along? I'm working on finding a way to relate Θ back to the velocity needed to get the ball back to the peg.
     
  16. Apr 13, 2014 #15

    haruspex

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    No, that's the speed needed to reach that height. But if the string is to stay taut it must be supplying (some nonzero portion of) the centripetal force. Consider the forces when in this position. What speed is it moving at if it's still moving in a circle but the string tension is just reaching zero.
     
  17. Apr 13, 2014 #16
    I got a ratio of L to r, but I would not swear it is correct. Is the answer given in the back of the book or something, to work towards? (actually looking at your thumbnail, I find I did a related (equivalent) problem, not this one)
     
  18. Apr 13, 2014 #17

    haruspex

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    r? What distance is that?
     
  19. Apr 13, 2014 #18
    I did the problem where it is released horizontally from height L, and hits a peg at L - r, so that r is the radius of the path. The mass does not make the orbit with radius r but the tension goes to zero at some angle. After the tension hits zero, the mass goes on a parabolic path to the peg. This may make the problem even harder.
     
  20. Apr 13, 2014 #19
    I got v=√(gL) for the tension just going to zero at the top.
     
  21. Apr 13, 2014 #20
    No, I have searched and searched. However, I did find a tarzan-physics problem that had to do with rotation and projectile motion. It seemed fairly complex though it may be helpful.

    http://arxiv.org/pdf/1208.4355.pdf
     

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