Question

A disc of radius r is cut out from a larger disc of radius 4r

Answer 1

Lets assume the centre of the bigger disc to be the origin.
The smaller disc is cut out from the bigger disc.
Radius of bigger disc = 2R
Radius of smaller disc = R
m₁=πR² x T x ρ          x₁=R         y₁=0
m₂=π(2R)² x T x ρ     x₂=0         y₂=0

where as ,T = Thickness of both discs
ρ = Density of both discs

Position of centre of mass = 

[(m₁x₁+m₂x₂)/(m₁+m₂) , (m₁y₁+m₂y₂)/(m₁+m₂)]

= [(-πR²TρR + 0) / (-πR²Tρ + π(2R)²TρR ) , 0 +0 / (m₁+m₂)] 

= [(-πR²TρR) / (3πR²Tρ) , 0 ]

= (-R/3 , 0) 

Centre of mass is at R/3 from the centre of bigger disc which is away from the centre of the hole.