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The problem you provided (Problem 2-24M) asks to determine the reactions for the cantilevered beam in Figure 2-62M. Let's break it down based on the provided diagram and information:

### Information from the diagram:
- **Beam type**: Cantilevered beam with a uniformly distributed load (UDL).
- **Distributed Load (w)**: 12 kN/m acting over a length of 6 m.
- **Cantilever support**: Located at point A.
  
### Reactions to determine:
Since the beam is cantilevered, the reactions at point **A** will include:
1. **Vertical reaction at A** ( \( R_A \) ),
2. **Moment reaction at A** ( \( M_A \) ),
3. **Horizontal reaction at A** ( \( H_A \) ).

However, in this case, no horizontal forces are present, so \( H_A = 0 \).

### Step-by-Step Solution:
1. **Vertical Reaction ( \( R_A \) )**:
   - The total load due to the UDL is \( w \times L \), where:
     \[
     w = 12 \text{ kN/m}, \quad L = 6 \text{ m}
     \]
     So, the total vertical load is:
     \[
     F = 12 \times 6 = 72 \text{ kN}
     \]
     This load acts at the center of the distributed load, which is at the midpoint of the beam (3 meters from point A).

   Therefore, the vertical reaction \( R_A \) must balance the total load:
   \[
   R_A = 72 \text{ kN}
   \]

2. **Moment Reaction at A ( \( M_A \) )**:
   - The moment is caused by the total distributed load acting at its center, which is 3 meters from point A.
     The moment is given by:
     \[
     M_A = F \times d = 72 \times 3 = 216 \text{ kN-m}
     \]

3. **Horizontal Reaction ( \( H_A \) )**:
   - Since there is no horizontal load in the problem, \( H_A = 0 \).

### Final Answer:
- \( R_A = 72 \text{ kN} \) (upward),
- \( M_A = 216 \text{ kN-m} \) (counterclockwise),
- \( H_A = 0 \text{ kN} \).

Let me know if you'd like more details or clarification on any of these steps!

### Relative Questions:
1. How do you compute reactions for beams with multiple distributed loads?
2. What is the difference between point loads and distributed loads in beam analysis?
3. How do moment diagrams for cantilevered beams typically look?
4. Can you explain how to find the centroid of complex distributed loads?
5. What happens to the moment if the distributed load is not uniform?

**Tip**: When dealing with distributed loads, always convert them to an equivalent point load acting at the centroid of the distribution.

Determine the reactions for the cantilevered beam of Figure 2-62M.

Determine the reactions for a cantilevered beam with a uniformly distributed load of 12 kN/m over a 6-meter span. This solution includes step-by-step calculations of the vertical reaction, moment reaction, and horizontal reaction. Suitable for undergraduate engineering students.

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