Energy and Center of Mass Problem

[bohobelle]

Homework Statement

Energy, Center of Mass: As a project your team is given the task of designing a space station consisting of four different habitats. Each habitat is an enclosed sphere containing all necessary life support and laboratory facilities. The masses of these habitats are 10 x 105 kg, 20 x 105 kg, 30 x 105 kg, and 40 x 105 kg. The entire station must spin so that the inhabitants will experience an artificial gravity. Your team has decided to arrange the habitats at corners of a square with 1.0 km sides. The axis of rotation will be perpendicular to the plane of the square and through the center of mass. To help decide if this plan is practical, you calculate how much energy would be necessary to set the space station spinning at 5.0 revolutions per minute. In your team's design, the size of each habitat is small compared to the size of the space between the habitats and the structure that holds the habitats together is much less massive than any single habitat.

Homework Equations

Krot = 1/2Iw^2
I = Ʃmiri

The Attempt at a Solution

I think that I'm supposed to start by finding the moment of intertia, but I'm not sure how to find that with such an odd shape. If someone could help me out, that would be great.
After that, I think that I'll plug it back into the Krot equation to solve for Krot and then find Emech? Honestly I have no idea what I'm doing!

 

[SteamKing]

1. Your equation for I is wrong.
2. The problem reads, "In your team's design, the size of each habitat is small compared to the size of the space between the habitats and the structure that holds the habitats together is much less massive than any single habitat." This means the inertia of each module about its own c.g. is negligible as is the structure holding the modules together. In other words, you have system composed of four particles.

 

[gneill]

Draw your square, placing a "habitat" at each corner. Determine the location of the center of mass, then the moment of inertia about that center.

Hint: Does the arrangement of the individual habitats (masses) matter?