With the given coordinates of points A, B, C, D, E, F based on the given dimensions. A(0, 0), B(27.52, 0), C(75.81, 0), D(0, 26.87), E(27.52, 26.87), F(75.81, 26.87) and center of gravity, which is Z(50.44, 10.13) ALL IN INCHES, show me the calculation of how the forces are calculated at each place to be hoisted upwards? AND THE OVERALL WEIGHT IS 400 LBS AND THE FORCES OF POINTS ARE NOT EVEN, EQUALLY DISTRUBUTED, WHAT IS THE LIFTING FORCES FOR EVERY POINTS?

To determine the lifting forces at each point \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) to hoist the object upwards, we need to perform a static equilibrium analysis, considering both the distribution of the weight and the geometry of the lifting points.

### Step 1: **Understanding the Problem**
- **Points** \(A(0, 0)\), \(B(27.52, 0)\), \(C(75.81, 0)\), \(D(0, 26.87)\), \(E(27.52, 26.87)\), and \(F(75.81, 26.87)\) represent the coordinates of the lifting points.
- The **center of gravity** is at \(Z(50.44, 10.13)\).
- The **total weight** of the object is 400 lbs.
- The forces at the lifting points are not equally distributed.

### Step 2: **Formulate the Problem**
The total weight of 400 lbs must be distributed among the six lifting points. The distribution of the forces depends on the relative positions of these points and the center of gravity. We assume that the object is in static equilibrium during the hoisting, so the sum of forces in both horizontal and vertical directions and the sum of moments about any point should be zero.

Let the forces at points \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) be \(F_A\), \(F_B\), \(F_C\), \(F_D\), \(F_E\), and \(F_F\) respectively.

### Step 3: **Set up the Equations**
1. **Sum of Vertical Forces:**
   \[
   F_A + F_B + F_C + F_D + F_E + F_F = 400 \text{ lbs}
   \]
   This equation ensures that the sum of the forces equals the total weight.

2. **Sum of Moments about the x-axis (considering moments around a horizontal axis passing through the center of gravity):**
   \[
   F_A \times 10.13 + F_B \times 10.13 + F_C \times 10.13 + F_D \times (10.13 - 26.87) + F_E \times (10.13 - 26.87) + F_F \times (10.13 - 26.87) = 0
   \]
   This equation ensures that the object does not rotate around the x-axis.

3. **Sum of Moments about the y-axis (considering moments around a vertical axis passing through the center of gravity):**
   \[
   F_A \times 50.44 + F_B \times (50.44 - 27.52) + F_C \times (50.44 - 75.81) + F_D \times 50.44 + F_E \times (50.44 - 27.52) + F_F \times (50.44 - 75.81) = 0
   \]
   This equation ensures that the object does not rotate around the y-axis.

### Step 4: **Solve the Equations**
Given that there are six variables (\(F_A\), \(F_B\), \(F_C\), \(F_D\), \(F_E\), \(F_F\)) and three equations, additional assumptions or information is needed to solve for each force. In practical scenarios, additional conditions or constraints (e.g., symmetry, specific force distribution) are often used to solve the problem.

However, the solution approach can involve setting up a system of equations and solving for the forces using matrix methods or optimization techniques to minimize the maximum force or distribute the forces as required.

### Step 5: **Example Calculation (Assuming Symmetric Distribution)**
If we assume the forces are symmetrically distributed about the center of gravity, the problem simplifies, and we can distribute the weight among the points. However, due to the asymmetry of the coordinates, we would still solve the system:

1. For simplicity, let's assume points \(A\) and \(B\) share the load equally, and similarly for the other pairs:
   \[
   F_A = F_D, \quad F_B = F_E, \quad F_C = F_F
   \]

2. We solve the system of three equations:
   \[
   2F_A + 2F_B + 2F_C = 400
   \]
   \[
   F_A \times 10.13 + F_B \times 10.13 + F_C \times 10.13 - F_A \times 16.74 - F_B \times 16.74 - F_C \times 16.74 = 0
   \]
   \[
   F_A \times 50.44 + F_B \times 22.92 + F_C \times -25.37 + F_A \times 50.44 + F_B \times 22.92 + F_C \times -25.37 = 0
   \]

   This system can be solved using numerical methods or symbolic computation tools.

### Conclusion
To find the precise values of \(F_A\), \(F_B\), \(F_C\), \(F_D\), \(F_E\), and \(F_F\), the system of equations must be solved numerically or symbolically. This provides the exact lifting force required at each point to hoist the object.

Would you like to go through the numerical solution in more detail, or is there another aspect you'd like to explore?

### Related Questions:
1. How would the forces change if the center of gravity was at a different location?
2. What would happen to the force distribution if the object were lifted at an angle?
3. How does the distance between the lifting points affect the force distribution?
4. Could we simplify the calculation by considering symmetry or another assumption?
5. How would the forces change if the object had a different total weight?

**Tip:** Always double-check the coordinates and ensure they align correctly with the object's geometry before performing static equilibrium calculations.

Learn how to calculate the lifting forces required at points A, B, C, D, E, and F to hoist an object upwards. The problem involves static equilibrium analysis using the coordinates of the points and the center of gravity. Suitable for undergraduate-level study in statics and equilibrium.

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