The image you've uploaded contains several mathematical physics problems related to kinematics and rolling motion. Here's a brief summary of the questions provided:

1. Condition for Rolling without Slipping:

   • A cylinder with radius $R_0$ rolls inside another hollow cylinder of radius $R_1$. We need to find the condition for rolling without slipping, relating the motion of the cylinders.

2. Movement of a Rod in a Cylinder:

   • A rod slides within a cylinder with radius $R_1$ and its ends, $A$ and $B$, remain in contact with the walls of the cylinder. This question asks to determine the movement of the point $G$ as a function of time.

3. Rotating Piece in a Plane:

   • A piece of material rotates in a plane with angular velocity. The goal here is to determine the velocity of a point on the rotating piece, specifically at the center $C$, and its position at a given time.

4. Rotation of a Plate:

   • A disk rotates about a horizontal axis. You are required to determine the velocities involved and express the conditions of motion for the disk.

Would you like help with solving one of these problems? Feel free to let me know which one, and I can guide you through the calculations or provide additional insights!