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Two reels rolling without slipping

  1. May 10, 2016 #1
    1. The problem statement, all variables and given/known data
    3q8Ftmn.jpg

    I have two cylindric reels laying on the ground. They are identical: mass M, composed by a cylinder of radius R and two side cylinders of radius 2R. I have to find angular acceleration and work done by F in delta t.

    2. The attempt at a solution

    On the first cylinder the torque is ##\tau_1=3RF-RT##, while on the second is ##\tau_2=RT##
    as the rope isn't extensible the two cylinders should have the same angular acceleration.
    So ##\alpha_1=\alpha_2##
    ##3RF-3RT=RT##
    ##T=-\frac{3}{2} F##

    ##\alpha=R/I *3/2 F=\frac{3RF}{2I}##
    Angular velocity of cylinder is ##\omega = \alpha t = \frac{3RF}{2I} t##.
    Kinetic energy of the first cylinder is ##1/2 I \omega^2= \frac{9R^2F^2t^2}{8I}## and this is also the work done by F plus the work done by T on the first cylinder. But as ##T/F=-3/2## and they are applied in the same way with respect to the displacement I expect ##W_T/W_F=-3/2## so ##W_F=2/5 \Delta K##. Is it correct?
    Thanks a lot
     
  2. jcsd
  3. May 10, 2016 #2

    BvU

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    Interesting, can you explain ?
    The rope can roll up (or off) on the reels, I presume ?
    [edit] A bit late, but I get it. Let me sit back and think a little longer before posting :nb)
     
  4. May 10, 2016 #3
    Oh my gosh, you're right.. I'm totally wrong(i don't know why but I considered it as a rod) and I don't know what to do now :/
     
  5. May 10, 2016 #4

    BvU

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    But the "##\tau_1=3RF−RT##, while on the second is ##\tau_2=RT##" is correct. That gives angular acceleration and therefore also (uniformly) accelerated translation
     
  6. May 10, 2016 #5
    I tried: (where A is friction force)
    ## F-A-T=M a_1 = M \alpha_1 2R ##
    ## T-A=M a_2 = M \alpha_2 2R ##

    Subtracting, I get: ##F-2T=M(2R)(\alpha_1-\alpha_2)##
    ##F-2T=M(2R)\frac{3RF-2RT}{I}##
    ##T=F\frac{1+6MR^2 / I}{2+4MR^2 / I}##

    Is it correct?
    What about the work? Is it correct my reasoning in applying kinetic energy theorem?
     
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