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. `(F)/(M) ( cos phi - sin phi) - mu g`
B. `(mu F)/(M)cos phi`
C. `(F)/(M) ( cos phi + sin phi) - mu g`
D. `(F)/(M)sin phi`

Explanation

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

### Step 1: Understand the Forces Acting on the Block
- The block experiences several forces:
- Weight \( W = mg \) acting downwards.
- Normal force \( N \) acting upwards.
- Applied force \( F \) at an angle \( \theta \) with the horizontal, which can be resolved into two components:
- Horizontal component: \( F \cos \theta \)
- Vertical component: \( F \sin \theta \)
- Frictional force \( f \) acting opposite to the direction of motion.

### Step 2: Determine the Normal Force
- The normal force \( N \) is affected by the vertical component of the applied force:
\[ N = mg - F \sin \theta \]

### Step 3: Express the Frictional Force
- The limiting friction \( f_L \) can be expressed in terms of the normal force:
\[ f_L = \mu N \]
- Given that \( \tan \phi = \mu \), we can write:
\[ f_L = \tan \phi \cdot N \]

### Step 4: Substitute for Normal Force
- Substitute the expression for \( N \) into the frictional force equation:
\[ f_L = \tan \phi \cdot (mg - F \sin \theta) \]

### Step 5: Set Up the Equation for Motion
- For the block to start moving, the horizontal component of the applied force must equal the limiting friction:
\[ F \cos \theta = f_L \]
- Substitute \( f_L \):
\[ F \cos \theta = \tan \phi \cdot (mg - F \sin \theta) \]

### Step 6: Rearranging the Equation
- Rearranging gives:
\[ F \cos \theta = \frac{\sin \phi}{\cos \phi} (mg - F \sin \theta) \]
- This simplifies to:
\[ F \cos \theta = mg \tan \phi - F \sin \theta \tan \phi \]

### Step 7: Collect Terms Involving \( F \)
- Collect all terms involving \( F \) on one side:
\[ F \cos \theta + F \sin \theta \tan \phi = mg \tan \phi \]
- Factor out \( F \):
\[ F (\cos \theta + \sin \theta \tan \phi) = mg \tan \phi \]

### Step 8: Solve for \( F \)
- Finally, solve for \( F \):
\[ F = \frac{mg \tan \phi}{\cos \theta + \sin \theta \tan \phi} \]

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