Roll Without Slipping

There are three forces:

$\text{Gravity parallel to ramp} = M g \, sin \theta \\ \text{Normal reaction} = M g \, cos \theta \\ \text{Critical Friction force} = \mu \, M g \, cos \theta \$

The sphere moment of inertia is $I = \frac{2}{5} M R^2$ . Examine the critical condition in which the maximum possible friction force is required. For rolling without slipping (suppose $\alpha$ is the rotation angle of the sphere about its own central axis):

$a_{CM} = R \, \ddot{\alpha} = R \, \frac{\tau}{I} \\ \frac{M g \, sin \theta - \mu \, M g \, cos \theta}{M} = R \, \frac{R \, \mu \, M g \, cos \theta }{\frac{2}{5} M R^2} \\ sin \theta - \mu \, cos \theta = \frac{5}{2} \mu \, cos \theta \\ sin \theta = \frac{7}{2} \mu \, cos \theta \\ \frac{2}{7} tan \theta = \mu$