Question

The radious of gyration about an axis through the centre of a hollow sphere with external radius a and internal radius b is

Answer 1

Solution:

MOI of the hollow sphere = MOI of a solid sphere of radius "a" - MOI of a solid sphere of radius "b"

let density of the sphere = d

M = Mass of sphere of radius "a" = d ×(4/3)πa^3

m = Mass of sphere of radius "b" = d ×(4/3)πb^3

So, MOI = (2/5)Ma^2 - (2/5)mb^2

= (2/5) (4/3) π d {a^5 - b^5}

since, MOI = mass of the shell × Radius of Gyration

=> (2/5)(4/3)πd (a^5 - b^5) = (M - m) R

=> R(4/3)πd (a^3 - b^3) = (2/5)(4/3)πd (a^5-b^5)

=> R = (2/5) ( a^5 - b^5)/(a^3 - b^3)