Question

Which of the following is true about the centre of mass of a body?
(A) Always lies at the geometrical centre
(B) Always lies inside the body
(C) Always lies outside the body
(D) Lies within or outside the body

Answer 1

Hint
 The centre of mass is a position defined relative to an object or system of objects; it is the average position of all the parts of the system, weighted according to their masses. Consider different examples and determine the location of the centre of mass in them to find the correct option.

Complete step by step answer
We know that, by definition, the centre of mass is a point where the weighted relative position of the distributed mass sums to zero. This means that it acts as the point where the entire mass of the body may be assumed to be concentrated to visualise its motion.
We know that:
1. Centre of mass can lie outside the body: The centre of mass of a uniform ring lies on its geometrical centre, which does not lie on the ring.
2. Centre of mass can lie on the body: The centre of mass of a uniform rod lies on the geometrical centre of the rod, which lies on the rod.
Hence, from 1 and 2, we can conclude that the centre of mass lies within or outside the body.
Option (D) is correct.

Note
The centre of mass for a uniform object always lies on the geometrical centre, but that is not the case for a non-uniform object. Consider the following square made of two materials: the left being a lighter metal and the right being a heavier metal, the centre of mass would lie on the right side which is not the geometrical centre of the square. Hence, the centre of mass does not always lie on the geometrical centre of the object.
The centre of mass is the particle equivalent of a given object. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. For a system of particles, the position R of the centre of mass can be given by,