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The problem asks to find the reactions at point \( O \) for a system with an external force applied at point \( B \) as shown in the diagram. 

Here is a step-by-step breakdown:

### Given Data:
- Force \( F = 80 \, \text{N} \) acting vertically upward at point \( B \), located at coordinates \( (6, 4) \).
- Points:
  - \( O(0,0) \): The point where reactions are to be calculated.
  - \( A(6,0) \): Point on the x-axis.
  - \( B(6,4) \): Point where the force is applied.

### Step 1: Free Body Diagram
We assume:
- \( R_x \) is the reaction force in the x-direction at point \( O \).
- \( R_y \) is the reaction force in the y-direction at point \( O \).
- \( M_O \) is the moment at point \( O \).

### Step 2: Equilibrium Equations
The system is in equilibrium, so the sum of forces and moments must be zero.

#### 1. Sum of forces in the \( x \)-direction (\( \Sigma F_x = 0 \)):
\[ R_x = 0 \]
Since there is no horizontal force acting on the structure, the reaction in the x-direction is zero.

#### 2. Sum of forces in the \( y \)-direction (\( \Sigma F_y = 0 \)):
\[ R_y - 80 = 0 \]
\[ R_y = 80 \, \text{N} \]
Thus, the vertical reaction at point \( O \) is \( 80 \, \text{N} \), balancing the external force applied at point \( B \).

#### 3. Sum of moments about point \( O \) (\( \Sigma M_O = 0 \)):
We take the moment of the applied force \( F \) about point \( O \). The force is acting at point \( B(6, 4) \), so the perpendicular distance from \( O \) to the line of action of the force is \( 6 \, \text{m} \).

The moment due to the force \( F \) is:
\[
M_O = 80 \times 6 = 480 \, \text{Nm}
\]
Since the force causes a counterclockwise rotation, the moment is positive.

Thus, the moment reaction at point \( O \) is:
\[
M_O = 480 \, \text{Nm}
\]

### Final Reactions at Point \( O \):
- \( R_x = 0 \, \text{N} \)
- \( R_y = 80 \, \text{N} \)
- \( M_O = 480 \, \text{Nm} \) (counterclockwise)

Would you like more details or have any questions?

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### Related Questions:
1. How would the reaction forces change if the force at point \( B \) were angled instead of vertical?
2. What happens to the reactions if the force applied at point \( B \) is doubled?
3. How do the coordinates of point \( B \) affect the reaction moments at point \( O \)?
4. How can you determine the reactions if there were additional horizontal forces at point \( B \)?
5. What is the role of moments in equilibrium analysis for statics problems?

### Tip:
Always use a consistent sign convention when summing moments and forces (e.g., counterclockwise moments are typically positive).

This solution provides a detailed step-by-step approach to finding the reactions at point O in a statics problem involving a force applied at point B. Concepts covered include the equilibrium equations, vector resolution, and the calculation of moments.

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