An object is supported on two scales with the scale readings

An object is supported on two scales, with the scale readings as shown. The mass moment of inertia about point A is 0.346 slug-ft².

Determine the mass moment of inertia about the center of mass, I_CM.
(include units with answer)

Solution

From parellel axis theorem

                                                     I = I_cm + M * d^2

           I = moment of inertia along the axis parallel to the axis passing thru the centre of mass and is located a distance d from the centre of mass

          I_cm = moment of inertia of the object along the axis passing thru the centre of mass

         M = mass of the object

        d = distance from the centre of mass to point A

       total mass of the object = 7.6 + 3.1 lb = 10.7 lb

assuming the centre of mass located at a distance d from A , hence the cm located at 3.0-d from other end

        imagine the object is just supported at its centre of mass , then the torque acting on both ends should cancel each other , and the object should stay in equilibrium with out being tilted to either end

                 if we balance the torques acting on either side ,

                            torce = gravity force * distance = Fg * l

                           =>    3.1 * (3.0 - d) = 7.6 * d

                                                                              => d = 0.86915

        the centre of mass is located at 0.86915 from point A

             using relation I = I_cm + M * d^2         

         I_cm = I - M * d^2    

the shape of the object was unknown , if it is rod then moment of inertia about centre of mass ML^2 / 12