Where is the center of mass for a sphere, cylinder, ring, and cube, and does it always lie within the object?

Here are the locations of the center of mass for each body, assuming uniform mass density, and the answer to your question:

• (i) Sphere: The center of mass is located at its geometric center.
• (ii) Cylinder: The center of mass is located at the midpoint of its axis of symmetry.
• (iii) Ring: The center of mass is located at the center of the ring (the center of the hollow space).
• (iv) Cube: The center of mass is located at its geometric center, where its diagonals intersect.

No, the center of mass of a body does not necessarily lie inside the body. A common example is a ring, where the center of mass is at the center of the hollow space, which is outside the material of the ring itself.