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To solve the given problem, we need to determine the reactions at the supports and the hinge. The structure is subjected to a point load \( F \) at point \( C \) and a uniformly distributed load \( q_0 \) along segment \( AB \). 

Let's outline the steps to solve this problem:

1. **Calculate the Equivalent Load for the Distributed Load:**
   The uniformly distributed load \( q_0 \) is acting over the length \( 3a \).

   \[
   w_{\text{total}} = q_0 \times 3a = 100 \, \text{N/mm} \times 300 \, \text{mm} = 30000 \, \text{N}
   \]

   The equivalent point load acts at the midpoint of \( AB \) (i.e., at \( 1.5a \) from \( A \)).

2. **Resolve the Point Load \( F \) into Components:**
   \[
   F_x = F \cos(\alpha) = 1000 \, \text{N} \cos(30^\circ) = 866 \, \text{N}
   \]
   \[
   F_y = F \sin(\alpha) = 1000 \, \text{N} \sin(30^\circ) = 500 \, \text{N}
   \]

3. **Determine the Reactions at Supports \( A \) and \( B \):**
   For equilibrium:
   - Sum of vertical forces \( \sum F_y = 0 \)
   - Sum of horizontal forces \( \sum F_x = 0 \)
   - Sum of moments about point \( A \) \( \sum M_A = 0 \)

   Let \( A_y \) be the vertical reaction at \( A \) and \( B_y \) be the vertical reaction at \( B \). \( A_x \) is the horizontal reaction at \( A \).

   - Horizontal force equilibrium:
     \[
     A_x - F_x = 0 \implies A_x = 866 \, \text{N}
     \]

   - Vertical force equilibrium:
     \[
     A_y + B_y - w_{\text{total}} - F_y = 0
     \]
     \[
     A_y + B_y = 30000 + 500 = 30500 \, \text{N}
     \]

   - Moment equilibrium about \( A \):
     \[
     B_y \cdot 3a - w_{\text{total}} \cdot 1.5a - F_y \cdot 4a = 0
     \]
     \[
     B_y \cdot 300 - 30000 \cdot 150 - 500 \cdot 400 = 0
     \]
     \[
     B_y \cdot 300 = 4500000 + 200000 = 4700000
     \]
     \[
     B_y = \frac{4700000}{300} = 15666.67 \, \text{N}
     \]

     Now, using the vertical force equilibrium equation:
     \[
     A_y + 15666.67 = 30500
     \]
     \[
     A_y = 30500 - 15666.67 = 14833.33 \, \text{N}
     \]

4. **Determine the Reaction at the Hinge \( G \):**
   The hinge \( G \) will have both vertical and horizontal reactions (\( G_x \) and \( G_y \)).

   - Horizontal equilibrium for segment \( BC \):
     \[
     G_x - A_x = 0 \implies G_x = 866 \, \text{N}
     \]

   - Vertical equilibrium for segment \( BC \):
     \[
     G_y - F_y = 0 \implies G_y = 500 \, \text{N}
     \]

### Summary of Reactions:
- \( A_x = 866 \, \text{N} \)
- \( A_y = 14833.33 \, \text{N} \)
- \( B_y = 15666.67 \, \text{N} \)
- \( G_x = 866 \, \text{N} \)
- \( G_y = 500 \, \text{N} \)

Would you like further details or have any specific questions regarding this problem?

Here are five questions you might consider:
1. How would the reactions change if the point load \( F \) were moved to a different location?
2. What if the angle \( \alpha \) were different?
3. How does the presence of the hinge \( G \) affect the reactions?
4. What are the shear force and bending moment diagrams for this beam?
5. How do the support reactions influence the design of the beam?

**Tip:** Always check the units and ensure consistency throughout the calculation to avoid errors.

Learn how to calculate support reactions and hinge forces in a structural analysis problem. This problem involves determining reactions at supports and a hinge under various loads, including point loads and distributed loads. Suitable for advanced engineering or physics students.

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