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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.

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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 figureFor m1:Ibb′=25m1r21and (Iaa′)1=Ibb′+m1(r1+12−a)2=25m1r21+m1(r1+12−a)2 (From parallel axis theorem)For m:ICC′=ml212,(Iaa′.)2=Icc′+ma2=ml212+ma2For m2:Idd′=25m2r22⇒(Iaa′)3=Idd′+m2(r1+12+a)2=25m2r22+m2(r1+12+a)2The moment of inertia of the given system about aa′Iaa′=(Iaa′)1+(Iaa′)2+(Iaa′)3={25m1r21(r1+12−a)2}+{25m2r22(r2+12−a)2}

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