Answer
$Case~3\lt Case~2\lt Case~1$
Work Step by Step
We know that
$I=\Sigma _i M_iR_i^2$
Case 1:
$I_x=(2M)(0)^2+2(\frac{M}{2})(2R)^2+2(3M)(R)^2$
This simplifies to:
$I_x=10MR^2$
Case 2:
$I_y=2(M)(R)^2+2(\frac{M}{2})(0)^2+2(3M)(R)^2$
This simplifies to:
$I_y=8MR^2$
Case 3:
$I_z=2(M)(R)^2+2(\frac{M}{2})(2R)^2+2(3M)(0)^2$
This simplifies to:
$I_z=6MR^2$
Thus, the ranking of the three cases in order of increasing moment of inertia is:
$Case~3\lt Case~2\lt Case~1$
