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Rolling sphere, where to start when i dont know the radius?

  1. Jun 4, 2014 #1
    Rolling sphere, problems with the fundementals.

    1. The problem statement, all variables and given/known data
    A sphere with the mass of 2.5 kg rolls without slipping. The speed of the center of gravity is 10 m/s.

    a) Calculate the translational kinetic energy of the sphere
    b) Calculate the rotational energy of the sphere
    c) Calculate the total kinetic energy of the sphere
    d) Repeat a)-c) for a hollow cylinder with the same mass and speed

    2. Relevant equations
    vT = ω x r ,where vT is the speed of the center of gravity for the sphere

    Er = 1/2 x I x ω^2 , where Er is the rotational energy and ω=angular velocity

    I = 2/5 x M x r^2 , moment of inertia of the ball

    Ek = 1/2 x m x vT^2 + 1/2 x I x ω^2 , where Ek is the total kinetic energy of the rolling motion


    3. The attempt at a solution

    My first thought was finding the radius using vT=ω x r , my book says that ω=2 rad/s

    r = (10 m/s)/(2 rad/s) = 10/2pi m, but im really not sure if im allowed to do this. I can't find any examples in my textbook that relates to this problem.
     
    Last edited: Jun 4, 2014
  2. jcsd
  3. Jun 4, 2014 #2
    Rotational energy is usually expressed as a function of ω. Using your Eqn. 2, it should be possible to express as a function of v. Now you can use ratios since the translational energy is known.

    Sorry I am somewhat cryptic but I don't want the hall monitors to come down on me.
     
  4. Jun 4, 2014 #3
    a)
    translational kinetic energy -> Ek = 1/2 x m x vT^2 = 1/2 x 2.5 kg x (10 m/s)^2 = 125 J

    b)
    rotational energy -> Er = 1/2 x I x ω^2 , I= 2/5m x r^2

    Er = 1/2 x 2/5 x m x r^2 x ω^2 , ω=vT / r

    Er = 1/5 x m x r^2 x (vT / r)^2

    Er = (1/5 x m x r^2 x vT^2)/r^2

    Er = 1/5 x m x vT^2 = 1/5 x 2.5 kg x (10 m/s)^2 = 50 J

    Am i somewhat close?
     
  5. Jun 4, 2014 #4

    tms

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    Is this value of ##\omega## given in the statement of the problem? Is anything else given that you haven't mentioned?
     
  6. Jun 4, 2014 #5
    This is the only information that is stated in the problem.
     
  7. Jun 4, 2014 #6

    tms

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    Where did the value for ##\omega## come from?
     
  8. Jun 4, 2014 #7
    I think I misinterpreted an example in my textbook, i haven't use that value in solving the problem above.
     
  9. Jun 4, 2014 #8

    tms

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    Do you mean you've solved it?
     
  10. Jun 4, 2014 #9
    To the best of my current understanding. My textbook have no right/wrong section so i have no way to verify my answer. It does not either have any examples that i find particulary useful helping my solving this problem.
     
  11. Jun 4, 2014 #10

    ehild

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    Yes, it is correct. Go ahead.

    ehild
     
  12. Jun 4, 2014 #11
    c) Total kinetic energy for the sphere:

    Ek = Ekt + Er = 125 J + 50 J = 175 J

    , is Ekt the correct notation for translational kinetic energy?
     
  13. Jun 4, 2014 #12

    ehild

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    You can use any notation if you explain what you mean. I would use KEt and KEr.
    And what about question d)?

    ehild
     
  14. Jun 4, 2014 #13
    Will complete it later today. Thank you so much for your help so far. Great site.
     
  15. Jun 4, 2014 #14

    ehild

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    Ok, I am looking forward to seeing your solution for d). :smile:

    ehild
     
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