The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρ$$Ar=krR$$ and ρ$$Br=krR5$$, respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centers are IA and IB, respectively. If $$IBIA=n10$$, the value of n is

Question

The densities of two solid spheres  A and B  of the same radii  R vary with radial distance  r as  ρ$$Ar=krR$$ and  ρ$$Br=krR5$$, respectively, where k  is a constant.  The moments of inertia of the individual spheres about axes passing through their centers are  IA and  IB, respectively.  If  $$IBIA=n10$$, the value of  n is

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At a distance of r, select a hollow sphere of thickness dr and mass dm. Moment of Inertia of this elemental Hollow sphere is given by, $$dl=23(dm)r2$$  here $$dm=$$ρ$$dV=$$ρ$$4$$$$π$$$$r2dr$$⇒$$dl=23$$ρ$$4$$$$π$$$$r2drr2$$ For  Sphere A:ρ$$=krRdl=23krR4$$$$π$$$$r2drr2$$⇒$$IA=8k$$$$π$$$$3R$$∫$$0R$$ $$r5dr$$⇒$$IA=4k$$$$π$$$$R59$$ For Sphere B:ρ$$=krR5dI=23kr5R54$$$$π$$$$r2drr2$$⇒$$IB=8k$$$$π$$$$3R5$$∫$$0R$$ $$r9dr$$⇒$$IB=4k$$$$π$$$$R515$$ Ratio $$=IBIA=4k$$$$π$$$$R5154k$$$$π$$$$R59=35=610$$ So, $$n=6$$

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