A uniform rod of mass 40 kg and length 3 m has moment of inertia about an axis AB passing through one end and perpendicular to the rod. A solid sphere of mass 10 kg and radius R has the same moment of inertia about an axis parallel to AB and located 3 m from the sphere's centre, as shown in the source figure. If R = √(α ÷ 2) m, find α.

Question

G1 Physics · Accepted Answer

Moment of inertia of the rod about an axis through one end and perpendicular to its length is I(rod) = ML² ÷ 3. Thus, I(rod) = 40×3² ÷ 3 = 120 kg·m². For the solid sphere, the moment of inertia about a diameter through its centre is I(CM) = (2 ÷ 5)mR². The required axis is parallel to this diameter and is 3 m away from the centre. By the parallel-axis theorem, I(sphere) = (2 ÷ 5)mR² + m(3)². With m = 10 kg, I(sphere) = (2 ÷ 5)(10)R² + 10×9 = 4R²+90. The two moments of inertia are equal: 120 = 4…