Principal Moment of Inertia-how to calculate?

[cccic]

Principal Moment of Inertia--how to calculate?

Homework Statement

I am writing a program that incorporates calculating the principal moment of inertia for a protein residue based on its component atom XYZ coordinates. I am exceedingly confused about which formulas to use in calculating principal moment of inertia for my situation.

Thus far my program does the following:

1. Calculate elements (Ixx, Iyy, Izz, etc.) of symmetric matrix A
2. Find the eigenvalues (and thus the principal moments of inertia)

Homework Equations

The formulas I originally used were from here: https://docs.google.com/viewer?a=v&q...ZFyxHNqw&pli=1

However looking at Wikipedia, the formulas for the symmetric matrix elements are different. http://en.wikipedia.org/wiki/Moment_...tia#Definition

I have also been scouring the internet and found open source code where the center of mass is subtracted from the x, y, and z coordinates before beginning calculation of elements Ixx, Iyy, Izz, etc.

The Attempt at a Solution

Which formulas/algorithm do I use to calculate principal moment of inertia for my case? I'm not sure where to begin with picking which formulas to use. Is there a source that is accessible to those with a weak physics background that will help me understand which formula/algorithm to use in calculating principal moment of inertia?

Thank you!

 

[tiny-tim]

hi cccic!

… However looking at Wikipedia, the formulas for the symmetric matrix elements are different. http://en.wikipedia.org/wiki/Moment_...tia#Definition

I have also been scouring the internet and found open source code where the center of mass is subtracted from the x, y, and z coordinates before beginning calculation of elements Ixx, Iyy, Izz, etc.

wikipedia is correct

(i can't compare it with your first link, since that isn't working )

wikipedia's formulas work for moment of inertia about any point

however, i expect you will usually need the moment of inertia about the centre of mass, so in that case yes you will have to subtract it from the x, y, and z coordinates first

 

[cccic]

This is the correct link I was referring to: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19640005930_1964005930.pdf

How do I find the center of mass for a protein residue that is composed of different atoms with individual x y z coordinates? Your answer seems to contend with another persons answer:

http://physics.stackexchange.com/qu...-to-calculate-the-principal-moment-of-inertia

 

[TSny]

Hi cccic. Ok, you have your molecule sitting in space. Imagine introducing an arbitrary Cartesian coordinate system with origin chosen at an arbitrary location and axes oriented arbitrarily. Then each atom in your molecule will have Cartesian coordinates (x,y,z) in this coordinate system. The equations given in Wikipedia will give you the elements of the moment of inertial tensor relative to the axes of your coordinate system. [The answer given to the question at the link to stackexchange makes an incorrect statement when it says that the formulas at Wikipedia are only for axes passing through the cm. They are in fact valid for any coordinate system.]

Now, suppose you want the moment of inertia tensor for a coordinate system that has origin at the center of mass of the molecule and with axes oriented parallel to your previously arbitrarily chosen coordinate system. Let's call this the center of mass coordinate system. Then you have two ways to go:

Method 1: Find the (x,y,z) coordinates of each atom in the center of mass coordinate system and then use the formulas for the moment of inertia tensor as given in Wikipedia using these coordinates. To find the coordinates of the atoms in the center of mass coordinate system, you would need to find the coordinates of the center of mass of the molecule (x_cm, y_cm, z_cm) in the original coordinate system using standard formulas (see http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html) Subtract these coordinates of the center of mass from each of your (x,y,z) coordinates of the atoms in the original coordinate system.

Method 2: Use the coordinates of the atoms in the original coordinate system, but use the formulas as given in the NASA document.

 

[paranormal]

I would suggest that you first start by understanding the concept of moment of inertia and how it relates to the physical properties of a system. This will help you determine which formulas and algorithms are most appropriate for your specific situation.

In general, the moment of inertia of a rigid body can be calculated using the formula I = ∫r²dm, where r is the distance from the axis of rotation and dm is the infinitesimal mass element. However, for a protein residue, which is a complex and irregularly shaped molecule, this approach may not be practical.

Instead, you can use the parallel axis theorem to calculate the moment of inertia for your protein residue. This theorem states that the moment of inertia of a body about any axis is equal to the moment of inertia about a parallel axis through the center of mass plus the product of the mass and the square of the distance between the two axes. This means that if you know the moment of inertia of your protein residue about an axis through its center of mass, you can use this theorem to calculate the moment of inertia about any other axis.

In your case, you can use the formulas from the Wikipedia link you provided, which are based on the parallel axis theorem, to calculate the moment of inertia about the principal axes. These formulas take into account the position and mass of each atom in the protein residue, making it more suitable for your application.

Additionally, it is important to note that the center of mass subtraction step in some of the open source code you found is likely used to simplify the calculation and may not be necessary for your specific situation. However, it is always a good practice to check the validity of your results and make sure they align with your expectations.

In summary, understanding the concept of moment of inertia and how it applies to your system, using the parallel axis theorem, and validating your results can help guide you in choosing the most appropriate formulas and algorithms for calculating the principal moment of inertia of your protein residue.