A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the disc coincide. The centre of mass of the new disc is `alphaR` from the centre of the bigger disc. The value of `alpha` is :

Options

A. `(1)/(2)`
B. `(1)/(6)`
C. `(1)/(4)`
D. `(1)/(3)`

Answer

D

Explanation

Let .O. be the centre of bigger disc of radius `2R=AO` and O. be the centre of smaller disc which has been removed from the bigger and has a radius `R=O.A`.

Mass of bigger disc `M=pi(2R)^(2).t.rho=4piR^(2)trho`
Mass of smaller disc `m=piR^(2)trho=piR^(2)trho=(M)/(4)`
Mass of remaining portion = `M-m`
`=M-(M)/(4)=(3)/(4)M`
Let G be the centre of mass of remaining portion and OG be its distance from the centre of mass of the bigger disc. Then taking moments about .O., we get
`(3)/(4)MxxOG=(M)/(4)xxO.O`
or `(3)/(4)OG=(R)/(4)orOG=(R)/(3)`
`alpha.R=(R )/(3)`
or `alpha=(1)/(3)`