Two solid spheres (A and B) are made of metals of different densities `P(A)` and `P_(B)` respectively. If their masses are equal, the ratio of their moments of inertia `(I_(A)//I_(B)` about their respective diameter is

Options

A. `((P_(B))/(P_(A)))^(2//3)`
B. `((P_(A))/(P_(B)))^(2//3)`
C. `(P_(A))/(P_(B))`
D. `(P_(B))/(P_(A))`

Answer

(a)

Explanation

The masses of two solid spheres are equal
i.e.,`m_(A)=m_(B)`
`(4)/(3)piR_(A)^(3)P_(A)=(4)/(3)piR_(B)^(3)P_(B)`
`((R_(A))/(R_(B)))^(3)=((P_(B))/(P_(A)))`
`((R_(A))/(R_(B)))=((P_(B))/(P_(A)))^(1//3)`
The expression for moment of inertia of sphere is given as, `I=(2)/(5)mR^(2)`
Let,
`I_(A)=(2)/(5)m_(A)R_(A)^(2)`
`I_(B)=(2)/(5)m_(B)R_(B)^(2)`
The ratio of moment of inertia is given as,
`(I_(A))/(I_(B))=((2)/(5)m_(A)R_(A)^(2))/((2)/(5)m_(B)R_(B)^(2))`
`=(R_(A)^(2))/(R_(B)^(2))` `(because m_(A)m_(B))`
`((P_(B))/(P_(A)))^(1//3)` (because From equation(1)).