Question

A thin uniform rod of length 'L' and mass 'M' is bent at the middle point 'O' at an angle of 45° as shown in the figure. The moment of inertia of the system about an axis passing through 'O' and perpendicular to the plane of the bent rod, is

• A. \( \frac{ML^2}{12} \)
• B. \( \frac{ML^2}{24} \)
• C. \( \frac{ML^2}{3} \)
• D. \( \frac{ML^2}{6} \)

Hint:

For composite systems like bent rods, break them down into simpler components and calculate the moments of inertia for each part. Combine the results for the total moment of inertia.

Answer 1

The Correct Option is A

Step 1: Understanding the moment of inertia.
The moment of inertia for a bent rod about the axis passing through the midpoint and perpendicular to the plane of the bent rod is a combination of the moments of inertia of two segments of the rod. Each segment has its own moment of inertia based on the formula for a rod rotating about an axis through its end.
Step 2: Formula application.
For a uniform rod of length \( L \) and mass \( M \), the moment of inertia about the axis through its center is \( \frac{ML^2}{12} \). Since the rod is bent at the midpoint at a 45° angle, the moment of inertia of each half is calculated accordingly. The final result is \( \frac{ML^2}{12} \).
Step 3: Conclusion.
The correct answer is \( \frac{ML^2}{12} \), as calculated by applying the principle of moments of inertia for a bent rod.