Question

A thin uniform rod has mass \( M \) and length \( L \). The moment of inertia about an axis perpendicular to it and passing through the point at a distance \( \frac{L}{3} \) from one of its ends, will be

• A. \( \frac{ML^2}{12} \)
• B. \( \frac{7}{8} ML^2 \)
• C. \( \frac{ML^2}{9} \)
• D. \( \frac{ML^2}{3} \)

Hint:

The parallel axis theorem allows you to calculate the moment of inertia when the axis of rotation is shifted from the center of mass.

Answer 1

The Correct Option is C

Step 1: Moment of inertia formula.
For a thin uniform rod, the moment of inertia about an axis perpendicular to it and passing through one of its ends is: \[ I = \frac{1}{3} ML^2 \] However, the axis of rotation is at a distance \( \frac{L}{3} \) from the end, not the center.
Step 2: Use of parallel axis theorem.
Using the parallel axis theorem, the moment of inertia about an axis a distance \( d \) from the center is: \[ I = I_{\text{center}} + Md^2 \] For this case, \( d = \frac{L}{3} \), so: \[ I = \frac{1}{12} ML^2 + M \left(\frac{L}{3}\right)^2 = \frac{1}{12} ML^2 + \frac{1}{9} ML^2 = \frac{ML^2}{9} \]
Step 3: Conclusion.
The moment of inertia is \( \frac{ML^2}{9} \), so the correct answer is (C).