A boy of mass M is applying a horizontal force to slide a box of mass M on a rough horizontal surface. The the floor is `mu` and the between the box and the floor is `mu'`. In which of the following cases it is certainly not possible to slide the box?

Options

A. `multmu',MltM'`
B. `mugtmu'MltM'`
C. `multmu',MgtM'`
D. `mugtmu',MgtM'`

Answer

To solve the problem, we need to analyze the forces acting on both the boy and the box, and determine the conditions under which the box cannot be slid by the boy.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Boy and the Box**:
- The boy applies a horizontal force \( F \) to the box.
- The box experiences a frictional force \( F_{r'} \) opposing the motion due to the rough surface beneath it.
- The frictional force between the boy and the ground is \( F_{r} \).

2. **Frictional Forces**:
- The maximum static frictional force that can act on the box is given by:
\[ F_{r'} = \mu' \cdot M \cdot g \]
- The maximum static frictional force that can act on the boy is given by:
\[ F_{r} = \mu \cdot M \cdot g \]

3. **Condition for the Box to Not Slide**:
- The box will not slide if the force applied by the boy \( F \) is less than or equal to the frictional force acting on the box \( F_{r'} \):
\[ F \leq F_{r'} \]
- This means that if the frictional force between the boy and the ground is not enough to overcome the frictional force acting on the box, the box will not slide.

4. **Analyzing the Conditions**:
- For the box to not slide, we need:
\[ F \leq \mu' \cdot M \cdot g \]
- Given that the boy can only exert a force up to the maximum frictional force between him and the ground:
\[ F \leq \mu \cdot M \cdot g \]
- Therefore, for the box to certainly not slide, we need:
\[ \mu \cdot M \cdot g \leq \mu' \cdot M \cdot g \]
- This simplifies to:
\[ \mu \leq \mu' \]

5. **Conclusion**:
- The box will certainly not slide if the coefficient of friction between the boy and the ground \( \mu \) is less than or equal to the coefficient of friction between the box and the ground \( \mu' \). This means that if \( \mu < \mu' \), the box will not move.

### Final Answer:
The box cannot be slid if \( \mu < \mu' \).