in which fo the following cases the centre of mass of a rod is certainly not at its centre?
(A) The density continuously increases from left ot right
(B) The density continuously decreases from left to right
(C) The density decreases from left to right up to the centre and then increases
(D) The density increases from left to right up to the centre and then decreases
Select correct alternative.

Options

A. Only (A)
B. Only (A) and (B)
C. Only (B)
D. Only (C) and (D)

Answer

To determine in which cases the center of mass of a rod is certainly not at its center, we will analyze each option provided:

### Step-by-Step Solution:

1. **Understanding Center of Mass**:
The center of mass (CM) of a rod is at its geometric center only if the mass is uniformly distributed. If the mass distribution is non-uniform, the CM will shift towards the denser part of the rod.

2. **Case A**: Density continuously increases from left to right.
- As we move from the left end to the right end, the density increases. This means that the mass is more concentrated towards the right end.
- Therefore, the center of mass will be located to the right of the geometric center.
- **Conclusion**: The center of mass is certainly not at the center.

3. **Case B**: Density continuously decreases from left to right.
- Here, the density is higher at the left end and decreases towards the right end.
- This means that the mass is more concentrated towards the left end.
- Therefore, the center of mass will be located to the left of the geometric center.
- **Conclusion**: The center of mass is certainly not at the center.

4. **Case C**: Density decreases from left to right up to the center and then increases.
- In this case, the left half of the rod has decreasing density, while the right half has increasing density.
- The left side is less dense, and the right side is more dense.
- Depending on the specific distribution of mass, it is possible for the center of mass to still lie at the geometric center if the mass on the right compensates for the mass on the left.
- **Conclusion**: The center of mass could potentially be at the center.

5. **Case D**: Density increases from left to right up to the center and then decreases.
- Similar to Case C, the left half has increasing density and the right half has decreasing density.
- Again, depending on the specific distribution, the center of mass could still lie at the geometric center.
- **Conclusion**: The center of mass could potentially be at the center.

### Final Answer:
The correct alternatives where the center of mass is certainly not at the center are **(A) and (B)**.