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 (x_n, y_n), as in the figure. 

(a) Show that the torques on the particles about the z-axis can be written τ_n = −x_nm_ng. 

(b) Show that if the center of gravity is located at (x_CG, y_CG), 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 y_CG = y_CM and z_CG = z_CM.)

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