Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?

The center of mass (COM) of an object is a point where the entire mass of the object can be considered to be concentrated for the purposes of analyzing its motion. The location of the center of mass depends on the shape and distribution of mass within the object. Here are the locations of the center of mass for the objects you mentioned:

(i) Sphere:
For a uniform-density sphere, the center of mass is at the geometric center, which is also the center of the sphere. So, the center of mass of a sphere is located at its exact center.

(ii) Cylinder:
For a uniform-density cylinder, the center of mass is also at the geometric center. In the case of a solid cylinder (without any holes or hollow sections), the center of mass is located at the midpoint of the axis of the cylinder, equidistant from the two circular bases.

(iii) Ring:
For a uniform-density ring (a flat, circular object with a hole in the center), the center of mass is at the center of the ring, specifically at the midpoint of the axis perpendicular to the plane of the ring.

(iv) Cube:
For a uniform-density cube, the center of mass is at the geometric center of the cube. This means it is located at the point where the diagonals of the cube intersect.

Regarding whether the center of mass of a body necessarily lies inside the body, it depends on the shape and distribution of mass within the body. In many cases, especially for compact, solid objects like the ones mentioned above (sphere, cylinder, cube), the center of mass does indeed lie inside the body. However, for more complex shapes or objects with irregular distributions of mass, the center of mass may not necessarily be located inside the physical boundaries of the body. For example, an irregularly shaped object with a cavity or void could have its center of mass outside the body.

In summary, the center of mass of an object is determined by its shape and mass distribution, and while it often lies inside the body, it may not always do so, depending on the object's characteristics.