A machinist begins with a uniform solid aluminum disk of mass M=3.0kg and radius R=0.20m. A circular hole of radius 2R is drilled completely through the disk such that the center of the hole is located a distance 2R from the geometric center of the original disk, measured in the plane of the disk.
Assume all removed material has the same mass density as the remainder of the disk and that the thickness of the disk is uniform. The axis of rotation is perpendicular to the plane of the disk and passes through the geometric center of the original disk.
(a) Describe qualitatively how drilling an off-center hole, rather than a concentric hole, affects the rotational inertia of the modified disk about the stated axis. In your explanation, reference the distribution of mass relative to the axis.
(b) Without performing numeric calculations, explain why the rotational inertia of the modified disk must be greater than 21MR2−21(4M)(2R)2, the value that would be obtained by naïvely subtracting the concentric-hole contribution from the full-disk value.
(c) Identify which theorem or physical principle justifies adding an extra term to correct the naïve subtraction in part (b), and state the mathematical form of the additional term using appropriate variables for any distances involved.