Two block of masses `m_(1)` and `m_(2)` connected by a light spring rest on a horixontal plane. The cofficient of friction between the block and the surface is equal to `mu`. What minimum constant force has to be applied in the horizontal direction to the block of mass `m_(1)` in order to shift the other block?

Explanation

To solve the problem of determining the minimum constant force that needs to be applied to block \( m_1 \) in order to shift block \( m_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on Block \( m_2 \)**:
- The force exerted by the spring on block \( m_2 \) is \( F_s = kx \), where \( k \) is the spring constant and \( x \) is the extension of the spring.
- The frictional force acting on block \( m_2 \) is given by \( F_f = \mu m_2 g \), where \( \mu \) is the coefficient of friction and \( g \) is the acceleration due to gravity.

2. **Set Up the Condition for Motion**:
- For block \( m_2 \) to start moving, the force exerted by the spring must overcome the frictional force. Therefore, we have:
\[ kx \geq \mu m_2 g \]

3. **Determine the Force on Block \( m_1 \)**:
- When a force \( F \) is applied to block \( m_1 \), it must overcome both the frictional force acting on itself and the force exerted by the spring on block \( m_2 \). The frictional force acting on block \( m_1 \) is \( F_{f1} = \mu m_1 g \).
- Hence, the total force equation can be expressed as:
\[ F = F_{f1} + F_s \]
- Substituting the expressions for the forces, we get:
\[ F = \mu m_1 g + kx \]

4. **Substituting the Condition for Motion**:
- From the condition for motion derived in step 2, we can substitute \( kx \) with \( \mu m_2 g \):
\[ F = \mu m_1 g + \mu m_2 g \]

5. **Factor Out Common Terms**:
- We can factor out \( \mu g \):
\[ F = \mu g (m_1 + m_2) \]

6. **Conclusion**:
- The minimum constant force \( F \) that must be applied to block \( m_1 \) to shift block \( m_2 \) is:
\[ F = \mu g (m_1 + m_2) \]

### Final Answer:
The minimum constant force required to shift block \( m_2 \) is:
\[ F = \mu g (m_1 + m_2) \]