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

Options

A. ` ( pi ) / ( 12 ) MR^ 2 `
B. ` (( 4 ) / (3) - ( pi ) / ( 4 ) ) MR ^ 2 `
C. ` (( 4 ) /( 3) - (pi ) /( 6 )) MR^ 2 `
D. ` (( 8 ) /( 3) - (10pi ) / ( 16 ) ) MR^ 2 `

Answer

D

Explanation

If M is mass of the square plate before cutting the holes, then mass of portion of each hole, `m=(M)/(16R^(2))xxpiR^(2)=(Mpi)/(16)`
`therefore` Moment of inertia of remaining portion of about Z axis.
`l=l_("square")-4_("hole")`
`=M/12(16R^(2)+16R^(2))-4[(mR^(2))/(2)+m(sqrt2R)^(2)]`
`=M/12xx32R^(2)-10mR^(2)`
`=8/3MR^(2)-(10)/(16)MR^(2)pi`
`l=((8)/(3)-(10pi)/(16))MR^2`