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### Step 1: Understanding the Problem
We have a uniform thin bar of mass m = 6 \, \text{kg} and length L = 2.4 \, \text{m} that is bent to form an equilateral hexagon. We need to find the moment of inertia about an axis passing through the center of mass and perpendicular to the plane of the hexagon.
### Step 2: Geometry of the Hexagon
An equilateral hexagon can be divided into 6 equal triangles. Each triangle has a base equal to the length of the side of the hexagon, which is s = \frac{L}{6} = \frac{2.4}{6} = 0.4 \, \text{m} . The height of each triangle can be calculated using the formula for the height of an equilateral triangle:
\[
h = \frac{\sqrt{3}}{2} s = \frac{\sqrt{3}}{2} \times 0.4 = 0.3464 \, \text{m}
\]
### Step 3: Finding the Center of Mass
The center of mass of the hexagon lies at its geometric center. For a uniform thin bar bent into a hexagon, the center of mass is located at the intersection of the diagonals of the hexagon.
### Step 4: Moment of Inertia Calculation
The moment of inertia I of a thin bar about an axis perpendicular to its length and passing through its center is given by:
\[
I = \frac{1}{12} m L^2
\]
However, since we have a hexagon, we need to consider the contribution of all 6 sides. The moment of inertia of each side about the center of mass is:
\[
I_{\text{side}} = \frac{1}{12} m_{\text{side}} s^2
\]
where m_{\text{side}} is the mass of each side. The total mass of the hexagon is distributed equally among the 6 sides, so:
\[
m_{\text{side}} = \frac{m}{6} = \frac{6}{6} = 1 \, \text{kg}
\]
Thus, the moment of inertia for one side is:
\[
I_{\text{side}} = \frac{1}{12} \times 1 \times (0.4)^2 = \frac{1}{12} \times 1 \times 0.16 = \frac{0.16}{12} = 0.01333 \, \text{kg m}^2
\]
### Step 5: Total Moment of Inertia for the Hexagon
Since there are 6 sides, the total moment of inertia I_{\text{total}} is:
\[
I_{\text{total}} = 6 \times I_{\text{side}} = 6 \times 0.01333 = 0.08 \, \text{kg m}^2
\]
### Step 6: Final Answer
The moment of inertia about the axis passing through the center of mass and perpendicular to the plane of the hexagon is:
\[
I = 0.08 \, \text{kg m}^2 = 0.8 \times 10^{-1} \, \text{kg m}^2
\]
Thus, the final answer is **0.8**.
