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

Answer

C

Explanation

( c) Area mass density : `sigma = (M)/((16 R^2))`
Mass of each hole : `m_1 = sigma R^2 =(M)/((16 R^2)) pi R^2 = (pi M)/(16)`
Distance between centre of plate and centre of hole `x = (4Rsqrt(2))/(4) = R sqrt(2)`. Moment of inertia of one hole about `z` axis : `I_1 = (1)/(2) m_1 R^2 + m_1 x^2`. Moment of inertia of whole plate about `z` axis : `I = (M(4 R)^2)/(6)`
Required Moment of Inertia
=`I - 4 I_1 =[(8)/(3) -(10 pi)/(16)] MR^2`.