A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs touch. The centre of mass of the new disc is at a distance aR from the centre of the bigger disc. The value of a is

Options

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

Answer

B

Explanation

Let mass of entire disc = M
Mass per unit area = `(M)/(pi(2R)^(2))= (M)/(4piR^(2))`
Mass of removed disc of radius R ,
`M_(1) = (M)/(4piR^(2))xxR^(2)= (M)/(4)`
Mass of remaining disc
`implies M_(2)= M- (M)/(4)= (3M)/(4)`
Center of mass of removed disc is `C_(1)` and centre of mass of remaining new disc is `C_(2)` And centre of mass of combination of `M_(1) ` and `M_(2)` will be at C(0,0).
`implies (M_(1)x_(1)-M_(2)-x_(2))/(M_(1)-M_(2))=0`
`implies M_(1)x_(1)=M_(2)x_(2)`
`(M)/(4).R=(3M)/(4)(alphaR) " " implies alpha=(1)/(3)`