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## Homework Statement

A cylinder of mass M and radius r is at distance l

_{0}from the edge. mass m is hanging from a rope. no friction at the weightless pulley. the cylinder starts rotating without sliding.

What's the acceleration of the COM of the cylinder.

What's the velocity of the COM of the cylinder when it reaches the edge.

## Homework Equations

Torque and acceleration of a rigid body: ##M=I\alpha##

Moment of inertia of a solid cylinder: ##I_c=\frac{mr^2}{2}##

Shteiner's theorem: ##I_b=I_a+md^2##

## The Attempt at a Solution

Relation between angular and linear acceleration of COM:

$$2r\alpha=\dot{x_A}\;\rightarrow\; \alpha=\frac{\dot{x_A}}{2r}$$

Tension in rope is T:

$$\left\{ \begin{array}{l} mg-T=m\dot{x_A} \\ 2rT=(I_c+Mr^2)\frac{\dot{x_A}}{2r} \end{array}\right.$$

$$\rightarrow \dot {x_A}=\frac{4mgr^2}{I_c+(M+4m)r^2}$$

The acceleration of the COM: ##\dot{x_c}=\frac{1}{2}\dot{x_A}##

The velocity at the edge:

$$v^2=2\dot{x_c}l_0\;\rightarrow\; v^2=\dot{x_A}l_0$$