The rigid object shown is free to pivot about the point O Th

The rigid object shown is free to pivot about the point O: The center of the large solid disk and the center of the large hollow ring are each 1 m from the point O. The large solid disk and the large hollow ring each have radius 1/2 m. The lengths of the two connecting rods extending to the small mass are shown. All connecting rods shown are essentially massless and the small mass has negligible rotational inertia relative to its center. Compute the moment of inertia of this object corresponding to rotation about the point O.

Solution

MOI of solid disk about its own axis = mr^2 /2

= 10*0.5^2 /2

= 1.25 kg-m^2

By parallel axis theorem, MOI of disk about O = 1.25 + 10*1^2

= 11.25 kg-m^2

MOI of ring about is own axis = mr^2

= 5*0.5^2

= 1.25 kg-m^2

By parallel axis theorem, MOI of ring about O = 1.25 + 5*1^2

= 6.25 kg-m^2

Distance from point mass to O = sqrt [ (sqrt3 + 2cos30)^2 + (2sin30)^2] = sqrt 13

MOI of point mass about O = 1*(sqrt 13)^2

= 13 kg-m^2

Total MOI about O = 11.25 + 6.25 + 13

= 30.5 kg-m^2