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Rolling motion of sphere (weird initial condition?)

  1. Dec 27, 2011 #1
    1. The problem statement, all variables and given/known data
    A spherical billiard ball radius a and mass M is sitting on a flat surface. The coefficient of friction between the ball and table is u. The moment of inertia for the billiard ball is
    2Ma^2/5. The ball is given a sharp horizontal hit at the centre and it starts moving with linear velocity Vo. and angular velocity wo=0. Vo is positive.
    a, Which way is the point of contact moving? is it rolling or slipping?
    b, Draw a diagram showing the forces, to they increase or decrease linear velocity? increase or decrease angular velocity?

    2. Relevant equations
    Friction force = u*m*g
    Parallel axis theorem, I= Icm + ml^2 (l is distance from centre)
    Newtons second law, F=ma
    Torque=rXF=dL/dt, F is force L is angular momentum.



    3. The attempt at a solution
    a, point of contact moving leftwards (if ball is rolling to the right), ball is rolling.
    b, Friction decreases linear velocity and increases angular velocity.
    (However i believe there is an error in the question, regarding the wo=0 part. The angular frequency and linear velocity should decrease linearly together to 0. Therefore if angular velocity is 0 at t=0 then it immedietly spikes up and starts decreasing afterwards.

    Anyway, here is my attempt at the solution, however i believe the error lies in the question which is that wo=0 and that it does not ask for the acceleration at a specific point of the ball, which would change the solution.

    Any help appreciated!
    http://imageshack.us/f/163/lastquestion.jpg
    http://imageshack.us/photo/my-images/155/lastquestion2.jpg
    http://imageshack.us/f/163/lastquestion.jpg
    http://imageshack.us/f/155/lastquestion2.jpg/
     
    Last edited: Dec 27, 2011
  2. jcsd
  3. Dec 27, 2011 #2

    Doc Al

    User Avatar

    Staff: Mentor

    If the ball moves to the right, how is the point of contact moving to the left? How can the ball be rolling, if ω0 = 0?
    OK.
    Why do you think there's an error with having ω0 = 0?
     
  4. Dec 27, 2011 #3
    Because of the frictious force between the table and the sphere, the sphere will start rolling, because there will be a torque exerted on the sphere. The linear velocity of the sphere is related to its rolling motion.
    Therefore, I must have made a mistake thinking that the ball was rolling, but with the condition that wo=0, the ball is not rolling at all?

    Also about the point of contact is initially stationary but moves to the left?


    Thankful for further help, this question is a bit messy for me.
     
  5. Dec 27, 2011 #4

    Doc Al

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    Staff: Mentor

    Yes, the friction force will start the sphere rolling. But at first it's not rolling, just translating.
    Not always. In this case the sphere starts out with a linear velocity but no rolling at all.
    Correct.

    If the ball is translating to the right, the point of contact will move to the right.
     
  6. Dec 27, 2011 #5
    Ok, So what would be wrong in my solution then? My solution says that the linear velocity decreases linearly and the rotational velocity increases linearly. The thing that bothers me is I find it very weird that when the ball has no linear velocity, it has rotational velocity.
    Also, You said that the sphere will start rolling, but then you answered that it will not roll at all? Which one is it?

    Thank you for the help.
     
    Last edited: Dec 27, 2011
  7. Dec 27, 2011 #6
    So the only way that I can fix this at this point is if I immidietly after t=0 force the rotational velocity up so that it decreases linearly as well along with the translational velocity.

    This will allow the rolling motion and the translational motion to cease at the same time.

    If this is not the case, then the my current solution says that when the ball stops it translational velocity, it will stand still and spin.

    Thankful for further help.
     
  8. Dec 27, 2011 #7

    Doc Al

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    Staff: Mentor

    The way I understand it is this. Initially, the ball translates but does not rotate. The friction force slows the linear speed, while increasing the rotational speed. At some point, the relationship between the linear and rotational speeds will be just right and the ball will be rolling without slipping.
     
  9. Dec 27, 2011 #8
    Yep, thank you very much for you're help.
     
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