Write the condition for rolling without slipping from an inclined plane.

(N/A) For an object of mass $M$,radius $R$,and moment of inertia $I = kMR^2$ (where $k$ is a constant) to roll down an inclined plane of angle $\theta$ without slipping,the static friction $f$ must satisfy the condition $f \le \mu_s N$.
The acceleration of the object is given by $a = \frac{g \sin \theta}{1 + k}$.
The force of friction is given by $f = \frac{mg \sin \theta}{1 + \frac{MR^2}{I}} = \frac{mg \sin \theta}{1 + \frac{1}{k}}$.
The normal force is $N = mg \cos \theta$.
Substituting these into the inequality $f \le \mu_s N$:
$\frac{mg \sin \theta}{1 + \frac{1}{k}} \le \mu_s mg \cos \theta$.
Thus,the condition for rolling without slipping is $\mu_s \ge \frac{\tan \theta}{1 + \frac{1}{k}}$.