Question

The centre of mass of a body:
A. Lies always at the geometrical centre
B. Lies always inside the body
C. Lies always outside the body
D. May lie within or outside the body

Answer 1

Hint: Recall that the centre of mass of a body depends on the shape of the body and mass distribution of the body. To this end, think of where the centre of mass of hollow objects would lie, given their mass distribution is entirely circumferential.

Complete step-by-step solution
Let us begin by understanding what centre of mass is and on what factors it depends.
The centre of mass of a system is defined at the point at which the whole mass of the system appears to be concentrated. Any external force applied acts on this point. In case of a rigid body, the centre of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid.
The position of the centre of mass of a body thus depends on two factors:
Shape of the body
Distribution of mass
These are the deciding factors in determining whether the centre of mass lies inside the body or outside the body. Here, inside and outside mean that if the centre of mass lies where we can find the physical structure (matter) of the body, we say it is inside while if it lies within the space that the body is present in but not on its physical structure (matter) then we say it lies outside the body.
A solid rigid symmetric body will have its centre of mass within the body.
However, the centre of mass may not necessarily lie within the body when
the object loses its geometric symmetricity: as in the case of a boomerang, where the centre of mass lies outside owing to its uneven mass distribution due to its shape.
The object is hollow, or has its mass distributed entirely on its circumference: as in the case of a ring, or a bangle or a concentric circle. For these, the centre of mass will be at the centre, where there is no mass (matter) present.
Therefore, the correct choice would be D. May lie within or outside the body

Note: Do not get confused between centre of mass and centre of gravity. Centre of mass, as we have defined earlier, is the point at which the distribution of mass is equal in all directions whereas centre of gravity is the point at which the distribution of weight is equal in all directions. Centre of mass arises from the geometric behaviour of the object and hence does not depend on the gravitational field whereas centre of gravity arises from the physical behaviour of the object and depends on the gravitational field acting on it.