# Mass pulling another, rotating, mass

1. Jun 25, 2016

### Karol

1. The problem statement, all variables and given/known data

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. Jun 25, 2016

### haruspex

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.

3. Jun 26, 2016

### Karol

Thank you very much Haruspex

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