What is the maximum value of force, so that both the blocks will move together ?

Answer

To solve the problem of finding the maximum value of force \( F \) so that both blocks move together, we can follow these steps:

### Step 1: Identify the Given Values
We have two blocks with the following properties:
- Mass of block 1 (\( m_1 \)) = 2 kg
- Mass of block 2 (\( m_2 \)) = 3 kg
- Coefficient of friction for block 1 (\( \mu_1 \)) = 0.2
- Coefficient of friction for block 2 (\( \mu_2 \)) = 0.6
- Acceleration due to gravity (\( g \)) = 9.8 m/s² (assuming standard value)

### Step 2: Calculate the Maximum Static Friction Forces
The maximum static friction force for each block can be calculated using the formula:
\[ f_{\text{max}} = \mu \cdot m \cdot g \]

For block 1:
\[ f_{1} = \mu_1 \cdot m_1 \cdot g = 0.2 \cdot 2 \cdot 9.8 = 3.92 \, \text{N} \]

For block 2:
\[ f_{2} = \mu_2 \cdot m_2 \cdot g = 0.6 \cdot 3 \cdot 9.8 = 17.64 \, \text{N} \]

### Step 3: Set Up the Equations of Motion
When both blocks are moving together, the force \( F \) must overcome the friction acting on both blocks. The total friction force opposing the motion is the sum of the friction forces acting on both blocks:
\[ F_{\text{friction}} = f_{1} + f_{2} = 3.92 + 17.64 = 21.56 \, \text{N} \]

### Step 4: Apply Newton's Second Law
For both blocks to move together, the applied force \( F \) must equal the total friction force:
\[ F = f_{1} + f_{2} \]

### Step 5: Calculate the Maximum Force
Thus, the maximum force \( F \) that can be applied so that both blocks move together is:
\[ F = 21.56 \, \text{N} \]

### Conclusion
The maximum value of force \( F \) so that both blocks will move together is approximately **21.56 N**.

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