How can the moment of inertia of an object be calculated on an arbitrary axis?

In summary, it seems that when finding the inertia tensor of an object, it is usually more convenient to calculate it in a "convenient" coordinate system (such as at the center of mass) rather than 0 to the center of mass.
  • #1
TheCanadian
367
13
This seems to be a crucial detail that I just glossed over, but when finding the inertia tensor of an object, is the origin always situated at the object centre of mass?

For example: In the link (http://hepweb.ucsd.edu/ph110b/110b_notes/node26.html ), is it necessary to do the integral from -s/2 to s/2 in each dimension as opposed to 0 to s if finding the inertia through the CoM?

Also, how exactly can one find the moment of inertia of an object on an arbitrary axis? Referring back to the link, if one wanted to find the moment of inertia on an axis making 30 degrees with the horizontal (but still running through the CoM), how exactly could the inertia tensor be transformed to do this?
 
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  • #2
The moment of inertia that you need is the one about the pivot point - the point about which the object is rotated. It is usually convenient to calculate this by taking the momen of inertia about the center of mass and use the parallel axis theorem to move to the point you want.
 
  • #3
Your moment of inertia tensor is only true for where you center your origin. In most cases centering your coordinate system at the center of mass and calculating the tensor from there is complicated. The usual method for these problems is first calculating your inertia tensor in a "convenient" coordinate system (usually at the base of your solid) and then performing a simple matrix operation on your tensor to "move it" into the center of mass coordinate system. If you'd like me to show you how to do so I can, although most textbooks on the subject will have an example on it as well.
 
  • #4
Mercy said:
Your moment of inertia tensor is only true for where you center your origin. In most cases centering your coordinate system at the center of mass and calculating the tensor from there is complicated. The usual method for these problems is first calculating your inertia tensor in a "convenient" coordinate system (usually at the base of your solid) and then performing a simple matrix operation on your tensor to "move it" into the center of mass coordinate system. If you'd like me to show you how to do so I can, although most textbooks on the subject will have an example on it as well.

I think I got it. Although I would be happy to read any material you might have.
 

1. What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in rotational motion. It depends on an object's mass and distribution of mass about an axis of rotation.

2. How is moment of inertia calculated?

The moment of inertia is calculated by integrating the mass of each infinitesimal element in an object with respect to its distance from the axis of rotation. The resulting formula is I = ∫r²dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the mass of the infinitesimal element.

3. What are the units of moment of inertia?

The units of moment of inertia depend on the units used for mass and distance. In the SI system, the units are kilogram-meter squared (kg m²). In the English system, the units are slug-foot squared (slug ft²).

4. How does the moment of inertia affect an object's rotational motion?

The moment of inertia affects an object's rotational motion by determining its rotational acceleration. Objects with a larger moment of inertia require more torque to rotate at the same angular acceleration as objects with a smaller moment of inertia.

5. Can the moment of inertia change for an object?

Yes, the moment of inertia can change for an object if its mass or distribution of mass changes. For example, a figure skater can change their moment of inertia by extending their arms out or pulling them in while spinning. Additionally, the moment of inertia can change if the object's shape changes, such as a rod rotating about its end versus its center.

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