A solid spherical ball and a hollow spherical ball of two different materials of densities `p_(1)` and `p_(2)` respectively have same outer radii and same mass. What will be the ratio the moment of inertia (about an axis passing through the centre) of the hollow sphere to that of the solid sphere?

Options

A. `(p_(2))/(p_(1))(1-(p_(2))/(p_(1)))^(5/3)`
B. `(p_(2))/(p_(1))[1-(1-(p_(2))/(p_(1)))^(5/3)]`
C. `(p_(2))/(p_(1))(1-(p_(1))/(p_(2)))^(5/3)`
D. `(p_(2))/(p_(1))[1-(1-(p_(1))/(p_(2)))^(5/3)]`

Answer

D

Explanation

Given that `p_(1)` is the density of material of solid sphere and `p_(2)` is density of material of hollow sphere.
Also, Rsolid = Rhollow = R (say)
and `M_("solid") = M_("hollow")`
`M=4/3pip_(1)R^(3)`
`=4/3pip_(2)[R^(3)-R_("inner")^(3)]rArrR^(3)-R_("inner")^(3)=(p_(1))/(p_(2))R^(3)`
`rArr_("inner")^(3)=R^(3)[1-(p_(1))/(p_(2))]`
`rArrR_("inner")=R[1-(p_(1))/(p_(2))]^(1//3)` ..(i)
Now, ratio of their moment of inertia about the central axis will be
`(I_("hollow"))/(I_("solid"))=(4/3piR^(3)p_(2)xx2/5R^(2)-4/3piR_("inner")^(3)p_(2)xx2/5R_("inner")^(2))/(2/5xx4/3piR^(3)p_(1)xxR^(2))`
`=(p_(2))/(p_(1))[(R^(5)-R_("inner")^(5))/(R^(5))]=(p_(2))/(p_(1))[1-((R_("inner"))/R)^(5)]`
`(I_("hollow"))/(I_("solid"))=(p_(2))/(p_(1))[1-[1-(p_(1))/(p_(2))]^(5//3)]` [From equation (i)]