Why do different origins result in different principal moments of inertia?

In summary, the conversation discusses the concept of finding the principal moments of inertia with respect to different origins for an arbitrary configuration. The speaker mentions that they end up getting different values due to the different mass distribution with respect to different axes, but the directions of the principal axes remain the same. They then provide a specific example of a square with 4 equal masses, where the principal moments of inertia are calculated with respect to two different origins, resulting in different values. The speaker poses the question of why this occurs.
  • #1
Geocentric
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Homework Statement


When I try finding the principal moments of inertia with respect to different origins for any arbitrary configuration(assuming that the inertia tensor is diagonalized), I end up getting different values. Intuitively, this is quite acceptable because the mass distribution is different with respect to axes corresponding to different origins. But in both the cases, the directions of principal axes are the same which is what really matter. I would like to have the thoughts of you guys as well on the statements I have made.


Homework Equations





The Attempt at a Solution

 
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  • #2
Sorry, but I don't understand exactly what your question is? And does this relate to a particular homework-type problem? (If not, perhaps someone should ask a mentor to move it to one of the physics discussion forums)
 
  • #3
Let me be more specific. Consider 4 equal masses at the 4 corners of a square of side b. First I took one of the corners as the origin and found the principal moments of inertia to be Ixx=mb^2, Iyy=3mb^2, Izz=4mb^2 after solving the secular equation. Again, I found the principal moments of inertia but now with respect to the center of mass as origin as Ixx=mb^2, Iyy=mb^2, Izz=2mb^2. Now my question is, why do I get different values of principal moments of inertia?
 

1. What are principal moments of inertia?

Principal moments of inertia are measures of an object's resistance to rotation about its principal axes. They are represented by the three principal moments of inertia, which are the moments of inertia about the three mutually perpendicular axes that pass through the object's center of mass.

2. How are principal moments of inertia calculated?

Principal moments of inertia can be calculated using the object's mass distribution and its distance from the axis of rotation. The formula for calculating the moment of inertia is I = ∫r² dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the mass element. This integral is evaluated for each of the three principal axes to determine the three principal moments of inertia.

3. What is the significance of principal moments of inertia?

Principal moments of inertia are important in understanding an object's rotational dynamics. They determine how an object will behave when subjected to a torque and are used in calculations of angular acceleration and angular momentum. They also play a crucial role in determining an object's stability and its ability to resist changes in its rotational motion.

4. How do principal moments of inertia differ for different objects?

The principal moments of inertia vary based on an object's mass distribution. Objects with more concentrated mass have larger principal moments of inertia, while objects with more spread out mass have smaller principal moments of inertia. Additionally, the shape and orientation of the object can also affect its principal moments of inertia.

5. Can principal moments of inertia be changed?

Yes, the principal moments of inertia can be changed by altering an object's mass distribution or shape. For example, adding weight to one side of an object will increase its moment of inertia about that axis. Additionally, the principal moments of inertia can be changed by changing the object's orientation. However, the total moment of inertia of an object remains constant unless an external torque is applied.

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