Starting with the formula for the moment of inertia of a rod rotated around an axis through one end perpendicular to its l?engt?h (?I = M ? l2 / 3), prove that the moment of inertia of a rod rotated about an axis through its center perpendicular to its? leng?th is ?I? l2 / 12 . You will find the graphics in Figure 10.12 useful in visualizing these rotations.

Step-by-step solution Step 1 of 2 For the moment of inertia of the rod rotating about an axis through one end perpendicular to its length, Suppose a uniform rod, of mass and length So the mass per unit length of the rod is Take a small element on the rod in between a distance from the center. So the mass of the element is Hence, perpendicular distance of the element from t he center axis is x. So the moment of inertia of this element about the center axis is,