# New mass moment of inertia of an object after rotation

1. Apr 23, 2012

### ravanbak

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

I hope I can explain this properly:

1. Let's say you're given the mass moment of inertia of some non-uniform 3-dimensional object.

2. The moment is relative to coordinate system "CS1", with origin at the object's center of mass and about axes x, y and z.

3. Say there is a second coordinate system, "CS2", with a different origin from CS1, but with all axes parallel to those of CS1.

What I'm really interested in is the object's moment of inertia relative to CS2, which I understand can be calculated using the Parallel Axis Theorem.

However:

4. Now, say the object is rotated about its own center of mass (origin of CS1) by some angle, θ, about some arbitrary axis, X.

The object will now have a new mass moment of inertia relative to the coordinate system CS2.

THE QUESTION: is there a way to use the initial mass moment of inertia of the object and the angle and axis of rotation to calculate the new moment of inertia of the object relative to CS2?

2. Relevant equations

This is more of a concept question, so no equations.

3. The attempt at a solution

My idea is to treat the initial moment of inertia of the object as a 3D vector and apply the same rotation to the vector as was applied to the object. Then I would use the rotated value of the vector as the new moment of inertia of the object and use the Parallel Axis Theorem to calculate the moment in coordinate system CS2.

Would this give the correct result?

Thanks in advance for any help.

2. Apr 23, 2012

### tiny-tim

welcome to pf!

hi ravanbak! welcome to pf!
before i go any further, can i check …

you do know that moment of inertia isn't a vector? …

that the (mass) moment of inertia about any axis is derived from the inertia tensor? (or matrix), and that there are three principal moments of inertia corresponding to the three eigenvectors?

3. Apr 23, 2012

### ravanbak

Re: welcome to pf!

Thanks for the reply. Yes, I do know it's not a vector and that there are 3 principal moments.

My thought was to treat those 3 moments as a single 3D vector, or a 3D point in space, and then apply the same rotation as was applied to the object to that point to get a new moment.

Just a thought experiment, not based on any formulas. Wouldn't be surprised if I'm completely wrong.

The problem that led to this question: you're given the moments on inertia of an object about a known x, y and z axis, but you know nothing else about the object. Now you need to rotate the object and figure out it's new moments of inertia about the original coordinate system axes.

4. Apr 23, 2012

### tiny-tim

not really following you

… is to use the inertia tensor in the usual way

5. Apr 23, 2012

### ravanbak

Sorry, I'm aware that I don't know enough to even properly ask the question. I'm going to do some reading about tensors now.

One more question: you said that the moment about an axis is derived from the inertia tensor. However, I don't have a tensor to begin with, only the 3 moments of inertia about the x, y and z axes.

Are you saying those 3 moments are enough to create the tensor?

6. Apr 23, 2012

### tiny-tim

if they are the 3 principal moments, yes, because then they're the diagonal values, and all the other values are zero