Newton's second law:

$\overline{){\mathbf{\Sigma}}{\mathbf{F}}{\mathbf{=}}{\mathbf{m}}{\mathbf{a}}}$

Relationship between angular acceleration, α, and linear acceleration, a:

$\overline{){\mathbf{a}}{\mathbf{=}}{\mathbf{r}}{\mathbf{\xb7}}{\mathbf{\alpha}}}$

Torque:

$\overline{){\mathbf{\tau}}{\mathbf{=}}{\mathbf{r}}{\mathbf{\xb7}}{\mathbf{F}}}$

Torque and angular acceleration:

$\overline{){\mathbf{\tau}}{\mathbf{=}}{\mathbf{I}}{\mathbf{\xb7}}{\mathbf{\alpha}}}$

Moment of inertia of a cylinder:

$\overline{){\mathit{I}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}{{\mathbf{r}}}^{{\mathbf{2}}}}$

where I is the moment of inertia.

A string is wrapped around a uniform solid cylinder of radius, r as shown in the figure (Figure 1). The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m. Note that the positive y direction is downward and counterclockwise torques are positive.

Find the magnitude α of the angular acceleration of the cylinder as the block descends.

Express your answer in terms of the cylinder's radius r and the magnitude of the acceleration due to gravity g.

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