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!

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!