In which of the following cases the centre of mass of a rod is certainly not at its centre

A) The density continuously increases from left to right

B) The density continuously decreases from left to right

C) The density decreases from left to right upto centre and then increases

D) The density increases from left to right upto centre and then decreases

When there is a non-uniform distribution of mass, the location with the highest mass has the mass centre. The centre of mass will not be at the centre of the supplied rod if the density continuously increases or decreases from right to left or from left to right.

The centre of mass of a rod is not at the middle, meaning that the mass of the rod varies with length. Also, the density is continuously altered depending on whether the rod is longer on the right or left.

Hence the correct answer is A & B.