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We need to calculate the minimum force \( F \) required to prevent block \( m_1 \) (23 kg) from slipping down while it is held against block \( m_2 \) (77 kg) on a frictionless surface.

### Step-by-Step Solution:

#### 1. **Forces on Block \( m_1 \)**:
   - The weight of block \( m_1 \) causes a downward force:
     \[
     W_1 = m_1 g = 23 \times 9.8 = 225.4 \, \text{N}
     \]
     where \( g \) is the acceleration due to gravity (9.8 m/s²).
   
   - The force of static friction between block \( m_1 \) and block \( m_2 \) resists this downward motion. The static frictional force \( f_s \) is given by:
     \[
     f_s = \mu_s N
     \]
     where \( \mu_s = 0.53 \) is the coefficient of static friction and \( N \) is the normal force exerted by block \( m_2 \) on block \( m_1 \).

#### 2. **Normal Force \( N \)**:
   The normal force \( N \) is provided by the horizontal force \( F \) pushing block \( m_1 \) against block \( m_2 \). Hence:
   \[
   N = F
   \]

#### 3. **Friction Condition**:
   To prevent block \( m_1 \) from slipping down, the static friction must balance the weight of block \( m_1 \):
   \[
   f_s \geq W_1
   \]
   Substituting for \( f_s \) and \( W_1 \):
   \[
   \mu_s F \geq 225.4
   \]
   Solving for \( F \):
   \[
   F \geq \frac{225.4}{0.53} = 425.28 \, \text{N}
   \]

### Conclusion:
The minimum force \( F \) required to hold block \( m_1 \) against block \( m_2 \) without slipping is **425.28 N**.

Let me know if you need further details or have any questions.

Here are some additional related questions:
1. What happens if the coefficient of friction is increased?
2. How does the mass of block \( m_2 \) affect the required force \( F \)?
3. What would happen if the surface beneath block \( m_2 \) were not frictionless?
4. How would the situation change if the direction of \( F \) was at an angle?
5. What would the required force be if \( m_1 \) had a different mass?

**Tip**: When solving friction-related problems, always identify the forces acting on the object and their direction to correctly apply the frictional force formula.

What is the minimum force F (in newtons) required to hold m1 against m2 (i.e., without m1 slipping downwards)?

Find the minimum force required to hold a 23 kg block against a 77 kg block without slipping, using static friction. Learn how to balance forces and apply the friction equation to solve this physics problem, suitable for high school students.

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