momentum of inertia is

Let analysis this problem from the contact point between the disk and the slope.

The normal force between the disk and the slope is , and the force along the slope is ,

Assume the friction between the disk and the slope is f.

The net force is , where a is the acceleration along the slope.

(the friction force from the slope to the disk is in the same direction as acceleration a)

The condition for rolling without slipping is

The torque is

So

Since the maximum static friction force f_{max}=mg\cos\theta\mu \ge mg\sin\theta,

it imply that for the disk to rolling without slipping.

If it is a ball instead of a disk, then .

So

.

So the condition for rolling without slipping becomes.