10.3 Exercise 21

Starting with the formula for the moment of inertia of a rod rotated around an axis through one end perpendicular to its length $\left(\right. I = Mℓ^{2} / 3 \left.\right)$ , prove that the moment of inertia of a rod rotated about an axis through its center perpendicular to its length is $I = Mℓ^{2} / 12$. You will find the graphics in the figure useful in visualizing these rotations.