# Cylinder pulled by rope on a rough surface

## Homework Statement

A rope is pulled off with a force F from a combined cylinder. the smaller cylinder's radius is r and the big one's is R. the cylinder rotates on a rough surface without sliding. the force makes an angle θ with the horizon.
What's the acceleration's "a" magnitude and direction.

## Homework Equations

Rigid body: ##M=I\alpha##
$$I=kMR^2$$

## The Attempt at a Solution

Torque relative to the contact point, the torque's arm is:
$$\left( R-\frac{r}{\cos\alpha} \right)\cos\alpha=R\cos\alpha-r$$
$$M=I\alpha\rightarrow F(R\cos\alpha-r)=kMR^2\cdot \alpha,\; R\alpha=a$$
$$\Rightarrow a=R\frac{F(R\cos\alpha-r)}{kMR^2}=\frac{F(R\cos\alpha-r)}{kMR}$$

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$$a=R\frac{F(R\cos\alpha-r)}{MR^2(1+k)}=\frac{F(R\cos\alpha-r)}{MR(1+k)}$$