A block of mass `m` is placed at rest on a horizontal rough surface with angle of friction `phi`. The block is pulled with a force `F` at an angle `theta` with the horizontal. The minimum value of `F` required to move the block is

Options

A. a. `(m g sin phi)/(cos (theta - phi))`
B. b. `(m g cos phi)/(cos (theta - phi))`
C. c. `mg tan phi`
D. d. `mg sin phi`

Explanation

To find the minimum value of the force \( F \) required to move a block of mass \( m \) placed on a horizontal rough surface with an angle of friction \( \phi \), we can follow these steps:

### Step 1: Draw the Free Body Diagram (FBD)
- Consider the block on the horizontal surface.
- The forces acting on the block are:
- The gravitational force \( mg \) acting downwards.
- The normal force \( N \) acting upwards.
- The applied force \( F \) at an angle \( \theta \) with the horizontal, which can be resolved into two components:
- Horizontal component: \( F \cos \theta \)
- Vertical component: \( F \sin \theta \)
- The frictional force \( f_r \) acting opposite to the direction of motion.

### Step 2: Set Up the Equations of Motion
- In the vertical direction, the forces must balance:
\[ N + F \sin \theta = mg \]
Rearranging gives:
\[ N = mg - F \sin \theta \quad \text{(Equation 1)} \]

- In the horizontal direction, the applied force must equal the frictional force:
\[ F \cos \theta = f_r \]
The frictional force can be expressed as:
\[ f_r = \mu N \]
where \( \mu = \tan \phi \). Substituting \( N \) from Equation 1 into this gives:
\[ f_r = \mu (mg - F \sin \theta) \]

### Step 3: Substitute and Rearrange
- Substitute \( f_r \) into the horizontal force equation:
\[ F \cos \theta = \mu (mg - F \sin \theta) \]
Expanding this gives:
\[ F \cos \theta = \mu mg - \mu F \sin \theta \]

### Step 4: Collect Like Terms
- Rearranging the equation:
\[ F \cos \theta + \mu F \sin \theta = \mu mg \]
Factor out \( F \):
\[ F (\cos \theta + \mu \sin \theta) = \mu mg \]

### Step 5: Solve for \( F \)
- Finally, solving for \( F \):
\[ F = \frac{\mu mg}{\cos \theta + \mu \sin \theta} \]

### Step 6: Substitute \( \mu \)
- Since \( \mu = \tan \phi \):
\[ F = \frac{\tan \phi \cdot mg}{\cos \theta + \tan \phi \sin \theta} \]

### Final Result
Thus, the minimum value of \( F \) required to move the block is:
\[ F = \frac{mg \tan \phi}{\cos \theta + \tan \phi \sin \theta} \]