A block A kept on an inclined surface just begins to slide if the inclination is `30^0`. The block is replaced by another block B and it is found that it just begins to slide if the inclination is `40^0`

Options

A. mass of A > mass of B
B. mass of A < mass of B
C. mass of A=mass of B
D. all the three are possible.

Answer

To solve the problem, we need to analyze the situation using the principles of friction and the forces acting on the blocks on an inclined plane.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Block A begins to slide at an angle of inclination of \(30^\circ\).
- Block B begins to slide at an angle of inclination of \(40^\circ\).
- We need to determine the relationship between the masses of blocks A and B.

2. **Identifying the Forces**:
- When a block is on an inclined surface, the forces acting on it include gravitational force and frictional force.
- The component of gravitational force acting down the incline is given by \(mg \sin \theta\), and the frictional force opposing this motion is given by \(f = \mu N\), where \(N\) is the normal force.

3. **Calculating the Normal Force**:
- The normal force \(N\) on an inclined plane is given by \(N = mg \cos \theta\).
- Thus, the frictional force can be expressed as \(f = \mu mg \cos \theta\).

4. **Setting Up the Condition for Sliding**:
- A block will begin to slide when the gravitational component down the incline equals the maximum static frictional force:
\[ mg \sin \theta = \mu mg \cos \theta \]
- This simplifies to:
\[ \tan \theta = \mu \]

5. **Applying the Condition for Both Blocks**:
- For block A at \(30^\circ\):
\[ \tan 30^\circ = \mu_A \quad \Rightarrow \quad \mu_A = \frac{1}{\sqrt{3}} \approx 0.577 \]
- For block B at \(40^\circ\):
\[ \tan 40^\circ = \mu_B \quad \Rightarrow \quad \mu_B \approx 0.839 \]

6. **Comparing the Coefficients of Friction**:
- Since \(\mu_B > \mu_A\), this indicates that block B has a greater coefficient of friction than block A.

7. **Conclusion on Mass Relationship**:
- The angle of inclination at which a block begins to slide is independent of its mass. Therefore, we cannot determine the mass relationship based solely on the angles at which the blocks slide.
- Thus, all three options regarding the mass of A and B are possible:
- Mass of A is greater than mass of B.
- Mass of A is less than mass of B.
- Mass of A equals mass of B.

8. **Final Answer**:
- The correct option is that all three conditions regarding the masses are possible.