A block of mass m is at rest on an inclined plane. The coefficient of static friction is `mu`. The maximum angle of incline before the block begins to slide down is :

Options

A. `"tan"^(-1)(mu//m)`
B. `"tan"^(-1)(mu)`
C. `"tan"^(-1)(mu//2)`
D. `"cos"^(-1)(mu)`

Answer

To solve the problem of finding the maximum angle of incline (θ) before a block of mass m begins to slide down, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Block**:
- The weight of the block (mg) acts vertically downward.
- This weight can be resolved into two components:
- Perpendicular to the incline: \( mg \cos \theta \)
- Parallel to the incline: \( mg \sin \theta \)

2. **Determine the Normal Force (N)**:
- The normal force (N) acting on the block is equal to the perpendicular component of the weight:
\[ N = mg \cos \theta \]

3. **Calculate the Frictional Force (F_f)**:
- The maximum static frictional force that can act on the block is given by:
\[ F_f = \mu N = \mu (mg \cos \theta) \]

4. **Set Up the Condition for Sliding**:
- The block will start to slide when the component of gravitational force parallel to the incline equals the maximum static frictional force:
\[ mg \sin \theta = \mu (mg \cos \theta) \]

5. **Simplify the Equation**:
- Cancel \( mg \) from both sides (assuming \( m \neq 0 \)):
\[ \sin \theta = \mu \cos \theta \]

6. **Divide Both Sides by \(\cos \theta\)**:
- This gives us:
\[ \tan \theta = \mu \]

7. **Find the Angle θ**:
- To find the angle θ, take the inverse tangent:
\[ \theta = \tan^{-1}(\mu) \]

### Final Answer:
The maximum angle of incline before the block begins to slide down is:
\[ \theta = \tan^{-1}(\mu) \]