A flat object in the xy -plane is free to rotate about the z -axis. The gravitational field is uniform in the y -direction.

A flat object in the xy -plane is free to rotate about the z -axis. The gravitational field is uniform in the ˆ’ y -direction. Think of the object as a large number of particles with masses mi located at coordinates (xi, yi), as in the figure?
(a) Show that the torques on the particles about the z -axis can be written ti = ˆ’ ximig.
(b) Show that if the center of gravity is located at (xCG, yCG), the total torque due to gravity on the object must be Σ(i = ˆ’ xCG Mg, where M is the total mass of the object.
(c) Show that xCG = xCM. (This same line of reasoning can be applied to objects that are not flat and to other axes of rotation to show that yCG = yCM and zCG = zCM.)

yi 2 Im2gI m4 Axis of rotation perpendicular to page