Minimum force F required to move the block

Options

A. `mu mg `
B. `(mu mg )/(2)`
C. `2 mu mg `
D. `(mu mg )/(4)`

Answer

To find the minimum force \( F \) required to move the block, we can follow these steps:

### Step 1: Understand the System
We have a block of mass \( m \) resting on a surface with a coefficient of friction \( \mu \). The block is connected to a pulley, and we need to determine the minimum force \( F \) applied to the pulley to move the block.

### Step 2: Draw a Free Body Diagram (FBD)
Draw the free body diagram for the block. The forces acting on the block include:
- The weight of the block \( mg \) acting downwards.
- The normal force \( N \) acting upwards.
- The frictional force \( f \) acting opposite to the direction of motion.

### Step 3: Identify the Normal Force
Since there is no vertical motion, the normal force \( N \) is equal to the weight of the block:
\[ N = mg \]

### Step 4: Calculate the Frictional Force
The frictional force \( f \) can be calculated using the formula:
\[ f = \mu N \]
Substituting for \( N \):
\[ f = \mu mg \]

### Step 5: Analyze the Pulley System
The force \( F \) applied to the pulley creates tension \( T \) in the string. For the block to start moving, the tension must overcome the frictional force. Therefore, we have:
\[ T = f \]
Thus,
\[ T = \mu mg \]

### Step 6: Relate the Force to Tension
In the case of the pulley system, the minimum force \( F \) required to move the block is related to the tension in the string. Since the pulley has two segments of the string supporting the block, we have:
\[ F = 2T \]
Substituting for \( T \):
\[ F = 2(\mu mg) \]

### Step 7: Final Expression
Thus, the minimum force \( F \) required to move the block is:
\[ F = 2\mu mg \]

### Answer
The minimum force required to move the block is \( F = 2\mu mg \).

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