Answer
$I = 1.8\times 10^{-4}~kg~m^2$
Work Step by Step
We can find the distance $r$ between each point and the z axis:
$r_1 = \sqrt{(2.0~cm)^2+(2.0~cm)^2} = 2.83~cm$
$r_2 = \sqrt{(0)^2+(4.0~cm)^2} = 4.0~cm$
$r_3 = \sqrt{(-3.0~cm)^2+(-3.0~cm)^2} = 4.24~cm$
$r_4 = \sqrt{(-2.0~cm)^2+(4.0~cm)^2} = 4.47~cm$
We can find the rotational inertia about the z axis:
$I = (0.050~kg)(0.0283~m)^2+(0.025~kg)(0.040~m)^2+(0.025~kg)(0.0424~m)^2+(0.030~kg)(0.0447~m)^2$
$I = 1.8\times 10^{-4}~kg~m^2$
