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Ball sliding on 3d-inclined plane

  1. Feb 9, 2015 #1
    1. The problem statement, all variables and given/known data

    http://www.sumoware.com/images/temp/xzdmdjtlksnfhiqc.png [Broken]
    A solid ball rolls perfectly with initial velocity v0 in horizontal axis ( y-axis ) on an inclined plane with elevation angle Θ as the picture above shown. This ball moves turning due to the gravitational acceleration till it has traveled distance L in x axis when it's at the bottom of the plane.
    Determine the time (t) the ball needs to get to the bottom of the plane ! (The ball doesn't slip while rolling)

    2. Relevant equations
    Rotational dynamics equation and linear kinematics equation
    Or conservation of energy equation

    3. The attempt at a solution

    I have two methods to solve the problem. But, the answers are different.

    Using rot. dynamics equation and linear kinematics.
    I just consider the x-axis since it's what the question asks.
    ΣFx = ma
    mg sin Θ - f = ma (Note : f is friction force)
    f = mg sin Θ - ma

    Στ = I α
    f R = I α
    f R = I (a/R)
    (mg sin Θ - ma) R = (2/5) m R^2 (a/R)
    mg sin Θ - ma = (2/5) m a
    g sin Θ - a = (2/5) a
    (7/5) a = g sin Θ
    a = (5/7) g sin Θ

    Then, I use the kinematics equation
    L = 0.5 a t^2
    2L/a = t^2
    14L/ (5g sin Θ) = t^2
    t = √(14L/5g sinΘ)

    But, using conservation of energy, I get different answer
    m g sin Θ L = (1/2) m v^2 + (1/2) I ω^2
    m g sin Θ L = (1/2) m v^2 + (1/2) (2/5 m R^2) (v^2 / R^2)
    g sin Θ L = (1/2) v^2 + (1/5) m v^2
    g sin Θ L = (7/10) v^2
    v = √(10 g sin Θ L / 7 )

    Then, I use the kinematics
    vt = vox + a t
    √(10 g sin Θ L / 7 ) = 0 + g sin Θ t
    t = √(10L/7g sin Θ)


    Which one is correct? Why?
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Feb 9, 2015 #2
    Oops.. I think that I forgot that the acceleration is not g sin theta since there is friction.. ( I used a = g sin theta in using the kinematics )
    Problem Solved :)
     
    Last edited by a moderator: May 7, 2017
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