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Mass pulling another, rotating, mass

  1. Jun 25, 2016 #1
    1. The problem statement, all variables and given/known data
    Snap1.jpg

    The cylinder rolls without friction. what's m's acceleration, M's angular acceleration and the tension.

    2. Relevant equations
    Shteiner's theorem: ##I_c=I_{c.o.m.}+Mr^2##
    Torque and angular acceleration: ##M=I\alpha##
    Moment of inertia of a solid cylinder: ##I_{cen}=\frac{mr^2}{2}##

    3. The attempt at a solution
    I denote the center variables with c
    Kinematics (or geometry? which discipline is suitable for this relation):
    $$x_c=\omega r~~\rightarrow~~\dot x_c=\alpha r~~\rightarrow~~\alpha=\frac{\dot x_c}{r}$$
    $$\left\{ \begin{array}{l} mg-T=m\dot x_A~~\rightarrow~~T=m(g-\dot x_A)=m(g-2\dot x_c) \\ 2r\cdot T=I_B\frac{\dot x_c}{r} \end{array} \right.$$
    $$\rightarrow~\dot x_c=\frac{2r^2mg}{I_B+4mr^2}=\frac{2r^2mg}{I_c+Mr^2+4mr^2}=\frac{2r^2mg}{\frac{Mr^2}{2}+(M+4m)r^2}=\frac{4mg}{3M+8m}$$
     
  2. jcsd
  3. Jun 25, 2016 #2

    haruspex

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    I assume you meant "rolls without slipping".
    Your use of x for velocity instead of displacement is unusual, but quite OK.
    Your working and answer look right to me.
     
  4. Jun 26, 2016 #3
    Thank you very much Haruspex
     
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