Consider a thin rigid sheet of material lying in the ry-plane. It has a form a square with side length 2a with the small

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Consider A Thin Rigid Sheet Of Material Lying In The Ry Plane It Has A Form A Square With Side Length 2a With The Small 1 (104.81 KiB) Viewed 55 times

Consider a thin rigid sheet of material lying in the ry-plane. 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 sketch below. The figure has uniform density (mass per area) o and a total mass M = 3a²o. P 0 Figure 1: A thin square of side 2a in the ry-plane with a square hole of side a at the center. (a) (8 points) With the origin at the center of mass of the object, find all components of the moment of inertia tensor in terms of a and M (please eliminate σ). (b) (4 points) With the origin at the point P shown in the figure, find all components of the moment of inertia tensor in terms of a and M. (e) (4 points) Imagine that the sheet is fixed to a shaft passing through the point P that rotates with constant angular velocity (not aligned along any particular axis). Tak- ing the origin at point P, consider the time dependence of three vectors: the angular momentum vector I, the angular velocity, and a unit vector that always points per- pendicular to the sheet ê3. 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 (0), not aligned along any particular axis. Taking the origin at the center of mass, consider the time t dependence of three vectors: the angular momentum vector L, the angular velocity vector (t), and a unit vector which always points perpendicular to the sheet ê3. State which of these vectors is constant in the space frame and which is constant in the body frame.