Four holes of radius R are cut from a thin square plate of side 4R and mass M. The moment of inertia of the remaining portion about z-axis is

Given : Radius of each circular hole=R
Side length of the square sheet L=4R
Moment of inertia of the complete sheet about z-axis Iz=ML²/6=M(4R)²/6=8/3MR²
Mass of the complete sheet is given as M
Area of one circular hole A=πR²
∴Mass of one circular hole m1=M(4R)² × πR²=πM/16
Moment of inertia of the hole about its center i.e O, Io=1/2m1R²=1/2 × πM/16R²=π/32MR²
∴Moment of inertia of one hole about z axis, I1Z=π/32MR² + m1d²
where d=OZ=√2R
⇒I1Z=π/32MR² + πM/16(√2R)²=5π/32MR²
Thus moment of inertia about z axis due to 4 circular holes ITZ=4 × I1Z=10π/16MR²
Net moment of inertia of the remaining sheet about z-axis, I'Z=IZ - ITZ
⇒I'Z=(8/3 - 10π/16)MR²=(8/3 - 5π/8)MR² = (64-15π)/24 MR²