## College Physics (4th Edition)

(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 $1.5~kg~m^2$ We can rank the three arrangements in increasing order of the rotational inertia: $b \lt a = c$
(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 $1.5~kg~m^2$ We can rank the three arrangements in increasing order of the rotational inertia: $b \lt a = c$