New mass moment of inertia of an object after rotation

In summary, the conversation discusses the calculation of an object's moment of inertia relative to a second coordinate system after being rotated about its own center of mass. The proposed solution involves treating the initial moment of inertia as a 3D vector and applying the same rotation to it, before using the Parallel Axis Theorem to calculate the moment in the new coordinate system. However, this approach may not be accurate as moment of inertia is not a vector and is derived from the inertia tensor. The conversation concludes with the suggestion to use the inertia tensor in the usual way, but notes that if the 3 principal moments of inertia are known, they can be used to create the tensor.
  • #1
ravanbak
9
0

Homework Statement



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?


Homework Equations



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


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.
 
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  • #2
welcome to pf!

hi ravanbak! welcome to pf! :smile:
ravanbak said:
… 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.…

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


tiny-tim said:
hi ravanbak! welcome to pf! :smile:


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?

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
not really following you :confused:

the answer to …
ravanbak said:
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.

… is to use the inertia tensor in the usual way
 
  • #5
tiny-tim said:
not really following you :confused:

the answer to …


… is to use the inertia tensor in the usual way

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
ravanbak said:
… 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?

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

(if they're not, then you need a lot more information)
 
  • #7
Thanks very much for your help, tiny-tim! I'm going to think more about this and possibly ask more questions later.
 

1. What is the definition of mass moment of inertia?

Mass moment of inertia is a measure of an object's resistance to rotational motion. It is a numerical value that describes how the mass of an object is distributed around its axis of rotation.

2. How does an object's mass moment of inertia change after rotation?

An object's mass moment of inertia changes after rotation due to the redistribution of its mass. The farther the mass is from the axis of rotation, the larger the moment of inertia will be.

3. What factors affect the mass moment of inertia of an object?

The mass moment of inertia of an object is affected by its mass, shape, and distribution of mass. Objects with larger mass or those with more mass farther away from the axis of rotation will have a larger moment of inertia.

4. How is mass moment of inertia calculated?

Mass moment of inertia can be calculated using the formula I = ∫r^2 dm, where r is the distance from the axis of rotation and dm is the infinitesimal mass element of the object.

5. Why is mass moment of inertia important in rotational motion?

Mass moment of inertia is important in rotational motion because it affects an object's ability to resist rotational acceleration. Objects with larger moments of inertia will require more torque to rotate at a given angular acceleration, and this property is crucial in various applications such as designing vehicles or machinery.

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