Question

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

• A. The density continuously increases from left to right.
• B. The density decreases from left to right up to the center and then increases.
• C. The density continuously decreases from left to right.
• D. The density increases from left to right up to the center and then decreases.

Answer 1

Answer: (A), (C)

The density continuously decreases from left to right.

Center of mass

1. The Centre of Mass of an object or a system of objects is a place defined relative to that object.
2. The Centre of Mass is the average position of all pieces that are weighted according to their Masses in a system.
3. The Centroid is the location of the Centre of Mass for simple rigid objects with homogeneous density.
4. It is possible that an object's Center of Mass does not always lie above it.
5. The Center of Mass of a uniform disc would be on its centre, but the Center of Mass of a ring would be on its centre but without any material.

The explanation of the correct option

In the case of option A,

1. The Center of mass of a rod is not at its centre of the rod i.e. mass of the rod varies according to length. Also, density continuously changes according to the length of the rod from left to right or right to left.
2. As the density continuously increases from left to right, there will be differences in the masses of rods that lie on either side of the centre of mass.
3. Thus, the centre of mass of a rod in such a case will certainly not be at its centre.

In the case of option C:

1. The Center of mass of a rod is not at its center of the rod i.e. mass of the rod varies according to length. Also, density continuously changes according to the length of the rod from left to right or right to left.
2. As the density continuously decreases from left to right, there will be differences in the masses of rods that lie on either side of the centre of mass.
3. Thus, the center of mass of a rod in such a case will certainly not be at its center.

The explanation of the incorrect option

In the case of option B:

1. The density decreases from left to right up to the centre and then increases.
2. Then the center of mass of the rod will be at its center.

In the case of option D,

1. The density increases from left to right up to the centre and then decreases.
2. Then the center of mass of the rod will be at its center.

Hence, options (A) and (C) are the correct options.