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 : .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`.

Four holes of radius `R` are cut from a thin square plate of side `4 R

1.	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 : .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» Correct Answer - C ( 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`.

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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 : .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`.

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