The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is ………….

Options

A. `1:sqrt(2)`
B. `1:3`
C. `2:1`
D. `sqrt(5):sqrt(6)`

Answer

D

Explanation

Circular Disc :

Moment of inertia about `y_(1)`
`I_(y_(1))=(MR^(2))/(4)`
According to parallel axis theorem,
Moment of inertia about `y._(1)`
`Iy._(1)=(MR^(2))/(4)+MR^(2)`
`therefore Iy._(1)=(5)/(4)MR^(2)`
also `Iy._(1)=Mk_(1)^(2)`
Circular ring :

`Iy_(2)=(MR^(2))/(2)`
According to parallel axis,
Moment of inertia about `y._(2)`
`Iy._(2)=(MR^(2))/(2)+MR^(2)`
`therefore Iy._(2)=(3)/(2)MR^(2)`
also `Iy._(2)=Mk_(2)^(2)`
`therefore (k_(1)^(2))/(k_(2)^(2))=(I_(y._(1)))/(I_(y._(2)))=((5)/(4)MR^(2))/((3)/(2)MR^(2))`
`therefore (k_(1)^(2))/(k_(2)^(2))=(5)/(6)`
`therefore k_(1):k_(2)=sqrt(5):sqrt(6)`