Answer
(a) The rotational inertia about the axis through masses B and C is $1.5~kg~m^2$
(b) The rotational inertia about the axis through masses A and C is $0.75~kg~m^2$
(c) The rotational inertia about the axis through the center is $1.5~kg~m^2$
Work Step by Step
(a) In general, $I = \sum M_i~R_i^2$, where $M_i$ is the mass of each point mass and $R_i$ is the distance from each point mass to the axis of rotation.
We can find the rotational inertia about the axis passing through masses B and C:
$I = (3.0~kg)(0.50~m)^2+(3.0~kg)(0)^2+(3.0~kg)(0)^2 + (3.0~kg)(0.50~m)^2$
$I = 1.5~kg~m^2$
The rotational inertia about the axis through masses B and C is $1.5~kg~m^2$
(b) We can find the rotational inertia about the axis passing through masses A and C:
$I = (3.0~kg)(0)^2+(3.0~kg)[(\sqrt{2})~(0.25~m)]^2+(3.0~kg)(0)^2 + (3.0~kg)[(\sqrt{2})~(0.25~m)]^2$
$I = 0.75~kg~m^2$
The rotational inertia about the axis through masses A and C is $0.75~kg~m^2$
(c) We can find the rotational inertia about the axis passing through the center:
$I = (3.0~kg)[(\sqrt{2})~(0.25~m)]^2+(3.0~kg)[(\sqrt{2})~(0.25~m)]^2 + (3.0~kg)[(\sqrt{2})~(0.25~m)]^2 + (3.0~kg)[(\sqrt{2})~(0.25~m)]^2$
$I = 1.5~kg~m^2$
The rotational inertia about the axis through the center is $1.5~kg~m^2$
