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Conveyor belt and friction- Determine slip time

  1. Sep 18, 2010 #1

    JJBladester

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    Gold Member

    1. The problem statement, all variables and given/known data
    A series of small packages is moved by a conveyor belt that passes

    over a 300mm radius pulley. The belt starts from rest at t=0 and its

    speed increases at a constant rate of 150 mm/s2. The

    coefficient of friction is μ=0.75. Determine the time the first

    package slips.

    Answer: ts=11.32s

    conveyor%20belt%20friction.jpg

    2. Relevant equations
    [tex]V_{0}=0[/tex]
    [tex]r_{pulley}=300mm[/tex]
    [tex]a_{pulley}=150mm/s^{2}[/tex]
    [tex]\mu=0.75[/tex]

    These are equations for polar coordinates, which I'm thinking I may need to use to solve the problem.
    [tex]F_{r}=ma_{r}[/tex]
    [tex]F_{\theta}=ma_{\theta}[/tex]
    [tex]F_{net}=F_{r}{\bf e}_{r}+F_{\theta}{\bf e}_{\theta}[/tex]

    [tex]m(\ddot{r}-r\dot{\theta}^{2})=F_{r}[/tex]
    [tex]m(r\ddot{\theta}+2\dot{r}\dot{\theta})=F_{\theta}[/tex]

    3. The attempt at a solution
    First of all, I'm trying to decide if I should use cartesian, polar, or path coordinates for this problem.

    If cartesian coordinates work, then I would set the problem up as below. Also, I don't think the radius of the pulley would matter...?

    Sum of forces x-dir = -Fs+Fpulley=ma
    Sum of forces y-dir = N-mg=0 -----> N=mg

    The maximum static friction would be Fssmg.

    For polar or path coordinates, I wouldn't know where to start.
     
  2. jcsd
  3. Sep 19, 2010 #2

    JJBladester

    User Avatar
    Gold Member

    I'm getting closer... I think.

    I actually think I can solve this without using polar or path coordinates.

    So, for constant linear acceleration, we have:

    vf2=vi2+2a(x2-x1)

    Because x2-x1 is really 2πr (arc length), we have:

    vf2=vi2+2a(2πr)

    I can solve this for vf and then plug that into vf=vi+at to get time... But I'm missing something with the whole Fssmg=ma... Like it should factor into the acceleration value for the two formulas above. Any help?
     
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