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Let's analyze and solve the problem based on the provided diagram and question details.

### Problem Analysis:
- A box of mass \( 24 \, \text{kg} \) is held in equilibrium in mid-air.
- It is supported by three rods exerting forces.
- The forces are described as follows:
  1. The force \((ai + aj) \, \text{N}\) acts diagonally upward to the right.
  2. The force \((-102i + 150j) \, \text{N}\) acts diagonally downward to the left.
  3. There is an unknown force exerted by the horizontal rod that we need to determine.

### Step 1: Identify the forces acting on the system.
- The mass of the box generates a downward gravitational force:
  \[
  F_g = m \cdot g = 24 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 235.2 \, \text{N} \, \text{downward (along the j-axis)}.
  \]
- Forces are in equilibrium, meaning the sum of forces in both the x-direction (i-axis) and y-direction (j-axis) must equal zero.

### Step 2: Set up equilibrium equations.
Let's break down the equilibrium conditions:
1. **Horizontal (x-axis) equilibrium:**
   \[
   ai - 102 + T = 0, \quad \text{(where T is the force in the horizontal rod)}
   \]
2. **Vertical (y-axis) equilibrium:**
   \[
   aj + 150 - 235.2 = 0.
   \]

### Step 3: Solve the vertical equilibrium equation.
From the vertical equation:
\[
aj + 150 = 235.2.
\]
\[
aj = 235.2 - 150 = 85.2.
\]
Thus:
\[
a = \frac{85.2}{j} = 85.2.
\]

### Step 4: Solve the horizontal equilibrium equation.
Now, substituting \( a = 85.2 \) into the horizontal equation:
\[
85.2 - 102 + T = 0.
\]
\[
T = 102 - 85.2 = 16.8 \, \text{N}.
\]

### Step 5: Determine if the force is tension or thrust.
- Since the value of \( T \) is positive, it acts along the direction of the rod, indicating that it is a thrust force.

### Final Answer:
The magnitude of the force in the horizontal rod is \( 16.8 \, \text{N} \), and it is a thrust.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How do we handle cases where forces are not perpendicular to each axis?
2. What changes if the box's mass or the direction of forces were altered?
3. How would the solution change if one of the rods was not rigid?
4. What are thrust and tension forces, and how can they be distinguished physically?
5. What role does the equilibrium condition play in real-world mechanical structures?

**Tip:** Always ensure to break forces into their components and apply equilibrium conditions in both the x and y directions to solve such problems efficiently.

A box of mass 24 kg is held in equilibrium in mid-air. It is supported by three light rigid rods, as shown in the diagram. All forces act in the vertical plane. The forces are (ai + aj) N and (-102i + 150j) N. Find the magnitude of the force in the horizontal rod and determine whether it is a tension or a thrust.

Learn how to solve an equilibrium problem where a 24 kg box is held in mid-air by three rods. This problem involves breaking forces into components, using Newton's laws, and determining the thrust force in the horizontal rod. Suitable for students in Grades 11-12.

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