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Question

Two uniform solid spheres of masses m1 and m2 and radii r1 and r2, respectively, are connected at the ends of a uniform rod of length l and mass m. Find the moment of inertia of the system about an axis perpendicular to the rod and passing through a point at a distance of a from the centre of mass of the rod as shown in figure.
981714_f55aad989a75405baf100ea80fc779f3.png

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Solution

Draw the axes bb,dd and cc passing through the centres of the spheres and the centre of mass of the rod as shown in figure
For m1:Ibb=25m1r21
and (Iaa)1=Ibb+m1(r1+12a)2=25m1r21+m1(r1+12a)2 (From parallel axis theorem)
For m:ICC=ml212,(Iaa.)2=Icc+ma2=ml212+ma2
For m2:Idd=25m2r22(Iaa)3=Idd+m2(r1+12+a)2=25m2r22+m2(r1+12+a)2
The moment of inertia of the given system about aa
Iaa=(Iaa)1+(Iaa)2+(Iaa)3={25m1r21(r1+12a)2}+{25m2r22(r2+12a)2}

1028776_981714_ans_2850cf9877b44b56a1c3dda0b86dba32.png

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