Censider a thin rigid sheet of material lying in the :ryplane. It has a form a square with side length 2a with the smaller central square with side length a cut out, as shown in the sloetch below. The gure has uniform density [mass per area} a and a total mass M = 31120. Figure 1: A thin square of side 2c in the :ryplane with a square hole of side a at the center. (a) [3 points} With the origin at the center of mass of the ohject1 nd all components of the moment of inertia tensor in terms of a and M [please eliminate J]. {b} (-4 points} 1With the origin at the point P shown in the gure1 nd all components of the n1oment of inertia tensor in terms of a and M. {c} [4 points} Imagine that the sheet is xed to a shaft passing through the point P that rotates with constant angular velocity fl [not aligned along anyr particular axis} Tak ing the origin at point P consider the time dependence of three vectors: the angular momentum vector L the angular velocityr , and a unit vector that always points per pudicular to the sheet e3. State which of these vectors is constant in the space frame and which is constant in the body frame. {d} [4 points} Now imagine that the sheet is rotating freely with no applied torques and with initial angular velocity Jim}, not aligned along any particular axis. Taking the origin at the center of mass, consider the time t dependence of three vectors: the angular mon1entum vector L, the angular velocity vector {Ii}, and a unit vector which always points perpendicular to the sheet as. State which of these vectors is constant in the space frame and which is constant in the hotlyr frame