A uniform sphere of mass `m` and radius R start rolling without slipping down an inclined plane. Find the time dependence of the angular. How will the result be affected in the case of a perfectly smooth inclined plane? The angle of inclination of the plane is `theta`.

Explanation

To solve the problem of a uniform sphere of mass \( m \) and radius \( R \) rolling down an inclined plane without slipping, we will analyze the forces acting on the sphere and derive the time dependence of its angular momentum.

### Step-by-Step Solution:

1. **Identify the Forces**:
On the inclined plane, the forces acting on the sphere are:
- Gravitational force \( mg \) acting downward.
- Normal force \( N \) acting perpendicular to the surface.
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