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?
(i) For a sphere, cylinder, ring, and cube, each having uniform mass density, the centre of mass coincides with their geometric centre.
(ii) No, the centre of mass of a body does not necessarily lie inside the body. For example, in the case of a ring with uniform mass density, the centre of mass is at its geometric centre, which lies in the hollow region where there is no mass.
Explanation
The textbook explains that for homogeneous bodies of regular shapes like rings, discs, spheres, etc., by using symmetry consideration, the centre of mass lies at their geometric centre. For every element dm at a point (x, y, z), there exists an element of the same mass at (-x, -y, -z), making the integrals zero. This establishes that the centre of mass coincides with the geometric centre. The ring example clearly demonstrates that the centre of mass can lie outside the physical material of the body, as the geometric centre of a ring is in the empty hollow space.