Moment of inertia tensor for two particles

[danbolin]

Hi,
I need to compute the inverse of the moment of inertia (MOI) tensor of a bunch of point particles in a simulation algorithm. The number and location of the particles differs at each evaluation. In all cases, I'm taking the particle coordinates with respect to their center of mass. Everything works fine for n > 2 (n being the number of particles), but the n = 2 case is problematic. Using the usual MOI tensor formulas (http://en.wikipedia.org/wiki/Moment_of_inertia#Moment_of_inertia_tensor ), one can in fact show that the resulting MOI tensor for two particles, where the positions are defined with the origin set to the center of mass of the two particles, is singular (zero determinant; the algebra is messy, but the proof is straightforward).

What am I doing wrong here? How can I get a non-singular MOI tensor for 2 particles?

 

[tiny-tim]

welcome to pf!

hi danbolin! welcome to pf!

I need to compute the inverse of the moment of inertia (MOI) tensor of a bunch of point particles in a simulation algorithm. The number and location of the particles differs at each evaluation.

i don't understand …

only rigid bodies have moments of inertia

what is the good of finding the moment of inertia of particles whose relative positions are changing? ​

 

[danbolin]

Hi tiny-tim,

Thanks for the reply!

I should have explained - this is all part of an explicit time integrator scheme, so that at each step, the particles can be considered (temporarily) rigid in order to calculate their velocities/positions at the next step. Although their relative positions change in time, at a particular point they can be considered a rigid body, and I need to compute their MOI tensor at that point. Does that make more sense?

In fact, let's forget the context, and simplify my question to this:

How does one properly calculate the MOI tensor for 2 particles? Why is it singular when the origin is taken to be the center of mass of the particles? The procedure works just fine for n > 2...

 

[tiny-tim]

… at each step, the particles can be considered (temporarily) rigid in order to calculate their velocities/positions at the next step. Although their relative positions change in time, at a particular point they can be considered a rigid body, and I need to compute their MOI tensor at that point. Does that make more sense?

not really

an unconnected system of particles has no common angular velocity, and without an angular velocity, a moment of inertia has nothing to act on

(even a pair of particles has no common angular velocity, unless their distance is constant)

 

[danbolin]

I probably made things more confusing with that explanation, sorry. Let's pretend they're connected particles that form a rigid body. What happens then for the case of n = 2, and how does one formulate a moment of inertia tensor? Why doesn't the same approach for n>=2 work?

 

[tiny-tim]

Let's pretend they're connected particles that form a rigid body. What happens then for the case of n = 2, and how does one formulate a moment of inertia tensor?

for n = 2, one of the principal moments of inertia (about the centre of mass) will be 0, and the other two will be 2m(L/2)²

 

[danbolin]

Thanks for your help!

One more quick question: would that mean that one of the eigenvalues of the MOI tensor would be 0, which makes the matrix singular? So it's actually impossible to come up with a MOI tensor that is invertible for n = 2?

 

[tiny-tim]

would that mean that one of the eigenvalues of the MOI tensor would be 0, which makes the matrix singular? So it's actually impossible to come up with a MOI tensor that is invertible for n = 2?

if the two particles are infinitely thin, yes

 

[danbolin]

Yup, sadly, they are infinitely thin..

Thanks again for your help!