Find the minimum value of horizontal force (F) required on the block of mass m to keep it at rest on the wall. Given the coefficient of friction between the surfaces is `mu`.

Answer

To solve the problem of finding the minimum horizontal force \( F \) required to keep a block of mass \( m \) at rest against a wall, given the coefficient of friction \( \mu \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Block**:
- The block is resting against a vertical wall, and the forces acting on it include:
- The gravitational force \( mg \) acting downward.
- The normal force \( N \) exerted by the wall on the block acting horizontally towards the left.
- The frictional force \( f \) acting upward, which opposes the gravitational force.

2. **Identify the Frictional Force**:
- The maximum static frictional force \( f \) can be expressed as:
\[ f = \mu N \]
- This frictional force must balance the weight of the block to keep it at rest. Therefore:
\[ f = mg \]

3. **Set Up the Equation**:
- Since the frictional force is equal to the weight of the block, we can substitute for \( f \):
\[ \mu N = mg \]

4. **Solve for Normal Force \( N \)**:
- Rearranging the equation gives:
\[ N = \frac{mg}{\mu} \]

5. **Relate Normal Force to Horizontal Force \( F \)**:
- The horizontal force \( F \) applied to the block is equal to the normal force \( N \) exerted by the wall:
\[ F = N \]

6. **Substitute \( N \) into the Equation for \( F \)**:
- Therefore, substituting the expression for \( N \) we found:
\[ F = \frac{mg}{\mu} \]

### Final Result:
The minimum value of the horizontal force \( F \) required to keep the block at rest on the wall is:
\[ F = \frac{mg}{\mu} \]