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The problem you shared involves calculating the moment of inertia for a composite object consisting of two uniform rods, each of mass \( M \) and length \( L \), about an axis passing through point \( O \). To solve this, we use the **parallel-axis theorem** and break the solution into parts for each rod, as the diagram indicates:

### Steps to Solve:

1. **Find the moment of inertia of rod A through point O:**
   - Rod A lies horizontally along the x-axis, with one end at \( O \). For a rod with mass \( M \) and length \( L \) rotating about one end, the moment of inertia is given by:
   \[
   I_A = \frac{1}{3} ML^2
   \]

2. **Find the moment of inertia of rod B through point O:**
   - Rod B is vertical and at a distance \( L \) from point \( O \), with its center at a distance of \( \frac{L}{2} \) from the axis. To apply the parallel-axis theorem, we first calculate the moment of inertia about its center and then shift it to \( O \):
     - Moment of inertia about the center of rod B:
     \[
     I_{\text{center}} = \frac{1}{12} ML^2
     \]
     - The parallel-axis theorem shifts this to point \( O \) (a distance \( \frac{L}{2} \)):
     \[
     I_B = I_{\text{center}} + M \left(\frac{L}{2}\right)^2 = \frac{1}{12} ML^2 + M \left(\frac{L}{2}\right)^2 = \frac{1}{12} ML^2 + \frac{1}{4} ML^2 = \frac{1}{3} ML^2
     \]

3. **Total moment of inertia:**
   The total moment of inertia is the sum of the moments of inertia of rods A and B:
   \[
   I_{\text{total}} = I_A + I_B = \frac{1}{3} ML^2 + \frac{1}{3} ML^2 = \frac{2}{3} ML^2
   \]

### Final Answer:
The total moment of inertia of the composite object about point \( O \) is:
\[
I_{\text{total}} = \frac{2}{3} ML^2
\]

Let me know if you need further clarification on any part of the solution!

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Here are 5 follow-up questions related to this topic:

1. What would the moment of inertia be if rod A were rotated about its center instead of the end?
2. How does the parallel-axis theorem apply if the axis of rotation is moved further away from rod B?
3. If the mass of rod B were different from rod A, how would the total moment of inertia change?
4. Can the moment of inertia for a system of rods change if the rods were connected at an angle instead of perpendicular?
5. How does increasing the length \( L \) of the rods affect the overall moment of inertia?

**Tip:** The parallel-axis theorem is crucial when calculating moments of inertia about non-central axes, so mastering it will greatly help in solving more complex rotational motion problems!

Find the moment of inertia of a composite object made of two similar uniform rods of mass M and length L about an axis passing through point O.

Solve the moment of inertia of a composite object consisting of two rods using the parallel-axis theorem. The solution covers calculating moments of inertia for both rods A and B, focusing on their orientations and applying key physics concepts. Perfect for university-level physics students.

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