# Moment of inertia with masses

1. Mar 27, 2013

### dinospamoni

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Ixx= ∫ y^2 + z^2 dm

Iyy= ∫ x^2 + z^2 dm

Izz= ∫ x^2 + y^2 dm

3. 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?

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2. Mar 27, 2013

### Staff: Mentor

They are points masses. Use sums instead of integrals.

3. Mar 27, 2013

### dinospamoni

Sorry, I'm not quite sure what you mean

4. Mar 27, 2013

### SteamKing

Staff Emeritus
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.

5. Mar 27, 2013

### 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?

6. Mar 27, 2013

### Staff: Mentor

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