Perpendicular-Axis Theorem Consider a rigid body that is a thin, plane sheet of arbitrary shape. Take the body to lie in the xy-plane and let the origin 0 of coordinates be located at any point within or outside the body. Let I. and I, be the moments of inertia about the x- and y-axes, and let 10 be the moment of inertia about an axis through 0 perpendiculars to the plane.
(a) By considering mass elements m, with coordinates (Xi' Yi)' show that/. + I, = 10, this is called the perpendicular-axis theorem. Note that point 0 does not have to be the center of mass_
(b) For a thin washer with mass M and with inner and outer radii R, and R'b use the perpendicular- axis theorem to find the momentum of inertia about an axis that is in the plane of the washer and that passes through its center. You may use the information in Table 9.2.
(c) Use the perpendicular axis theorem to show that for a thin, square sheet with mass M and side L, the moment of inertia about any axis in the plane of the sheet that passes through the center of the sheet is ML you may use the information in Table 9.2.