A flat object in the xy-plane is free to rotate about the z-axis. The gravitational field is uniform in the −y-direction. Think of the object as a large number of particles with masses mn located at coordinates (xn, yn), as in the figure.
(a) Show that the torques on the particles about the z-axis can be written τn = −xnmng.
(b) Show that if the center of gravity is located at (xCG, yCG), the total torque due to gravity on the object must be Στn = −xCGMg, 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.)
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