How Do You Calculate the Inertia Tensor for a Complex Molecule?

[dinospamoni]

Homework Statement

A molecule similar to methanol is made by joining three flourine atoms (purple; m = 19 amu each) to one carbon atom (blue; 12 amu) to one oxygen atom (green; 16 amu) to one potassium atom (orange; 39 amu). The position of each atom is as follows:

1. The flourine atoms are evenly spaced about the origin in the z = 0 plane, with one of them on the x-axis at -0.476 nm.

2. The carbon atom is on the z axis at z = 0.238 nm.

3. The oxygen atom is on the z axis at z = 0.952 nm.

4. The potassium atom is located at coordinates (x, y, z) = (-0.357, 0.000, 1.190) nm.

Determine the components of the inertia tensor for this molecule. Enter a) Ixx, b) Iyy, and c) Izz.

Picture is attached

Homework Equations

Ixx= ∫ y^2 + z^2 dm

Iyy= ∫ x^2 + z^2 dm

Izz= ∫ x^2 + y^2 dm

The Attempt at a Solution

I'm not sure where to start with this one. I've only ever had to find the moment of inertia for a solid object, never a collection of small objects. For those I converted to spherical coordinates and solved it that way.

I think the first thing that would help me out is finding out how to integrate over mass, or finding a substitute for it which would be easier to integrate over. Any suggestions?

 

[DrClaude]

They are points masses. Use sums instead of integrals.

 

[dinospamoni]

Sorry, I'm not quite sure what you mean

 

[SteamKing]

You can ignore the moment of inertia of each atom about its own centroidal axis. Apply the parallel axis theorem after finding the location of the centroid of the molecule.

 

[dinospamoni]

I think what you're saying is that a molecule only adds to the moment of inertia of it isn't lying directly on that axis?

ie; the blue and green molecules don't add to the Izz moment of inertia because they lie on the z-axis?

 

[DrClaude]

$$
I_{xx} = \sum_i m_i \left( y_i^2 + z_i^2 \right)
$$
etc. where ##y_i## is the ##y## coordinate of atom ##i## and so on.