Inertia tensor for point masses

[physicsdude101]

Homework Statement

Three equal point masses, mass M, are located at (a,0,0), (0, a, 2a) and (0, 2a, a). Find the centre of mass for this system. Use symmetry to determine the principle axes of the system and hence find the inertia tensor through the centre of mass. (based on Hand and Finch, Chapter 8 Problem 9).

Homework Equations

$$I_{xx}=\sum_{i} m_i(y_i^2+z_i^2)$$ ,$$I_{xy}=-\sum_{i} m_i x_i y_i$$ and $$\mathbf{R_{CM}}=\frac{\sum_{i} m_i\mathbf{r_i}}{\sum_{i} m_i}$$

The Attempt at a Solution

I got that the centre of mass was (a/3,a,a) but I'm not sure how to find the principle axes using symmetry as I can't really visualise the 3D setup that well.

 

[DrClaude]

I got that the centre of mass was (a/3,a,a) but I'm not sure how to find the principle axes using symmetry as I can't really visualise the 3D setup that well.

Make a drawing!

 

[physicsdude101]

Make a drawing!

I don't get it from the drawing I made either. Oops should've said I did one earlier.

 

[DrClaude]

The three points form a plane, and two of the three points are at equal distances from the center of mass. Both of these allow you to find two of the axes (the second one is a symmetry axis), and the third axis will be perpendicular to both.

 

[physicsdude101]

The three points form a plane, and two of the three points are at equal distances from the center of mass. Both of these allow you to find two of the axes (the second one is a symmetry axis), and the third axis will be perpendicular to both.

I think I worked it out: Do the principal axes point in the directions (-2,3,3),(3,1,1),(0,1,-1)?