The ratio of the radii of gyration of a circular disc and a circular ring of the same radii about a tangential axis perpendicular to plane of disc or ring is

Options

A. `(sqrt3)/(2)`
B. `2:3`
C. `1:2`
D. `sqrt5:sqrt6`

Answer

A

Explanation

For a disc, the M.I. about a tangential axis perpendicular to the plane of the disc is calculated by using the theorem of parallel axes.
`I=(1)/(2)mR^(2)+MR^(2)=(3)/(2)mR^(2)`
`because" "I=MK^(2)" "therefore" "K_("disc")=sqrt((I)/(M))=sqrt((3)/(2)R^(2))=sqrt((3)/(2))R`
and for the ring, for a similar axis,
`I=mR^(2)+mR^(2)=2mR^(2)`
`therefore" "K_("ring")=sqrt((I)/(M))=sqrt(2R^(2))=sqrt2R`
`therefore" "(K_("disc"))/(K_("ring"))=(sqrt((3)/(2)))/(sqrt2)=(sqrt3)/(2)`