Calculating Moment of Inertia (2D rectangles)

In summary, the moment of inertia for a 2D object is based on the object's centre of mass, which can be determined by dividing the object into two parts and calculating the respective moments of inertia.
  • #1
Nanako
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Hi everyone. I'm presently improving a 2D physics engine i wrote, with angular dynamics. Moment of inertia is becoming a big sticking point for me.

However, given that my applications are fairly simple, I'm hoping someone can help me come up with a simple way of calculating the moment of inertia for stuff.

All of my objects for now will (as far as physics is concerned) be rectangular, or made up of several rectangles. And i know there's a table of common moments of inertia that i might use, however that is mostly concerned with axes of rotation that pass through the centre of the object, or one of it's ends. i can't always guarantee that. I need to figure out a way of calculating for objects with an arbitrary centre of mass, which may not always be anywhere near the centre of geometryAlso another related question:

Take for example, a hammer.
if i divide it into two rectangles (the head, and the shaft) each of which has an evenly distributed mass over its area (but a different mass each) resulting in the centre of mass of the total object being closer to the head,

would that help to simplify the calculations involved, as opposed to:

making the whole hammer a single rectangle (with an unevenly distributed mass)I'd like to get this system working for rectangles, and then later i can add support for circles (and ovals). Rectangular and rounded shapes would probably cover all of my physics needs for the forseable future on this project. I don't have any need of calculating I for weird and irregular shapes, which i hope would make this simpler.

Any thoughts and references on this subject would be appreciated.
 
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  • #3
hi tim, thanks for the help!

I was aware of the PA theorem, but i had already excluded it as being useful to me, I'm not fully understanding how it relates to my needs here. This theorem seems to largely depend on already knowing I through the object's centre of mass, first.

it mentions 2d planes and the centroid, but it assumes even distribution of mass there, (and therefore, assumes that the centroid is also the centre of mass).

The latter part might be useful for my two-part hammer idea, although how do i combine I on two parts like the hammer example.

also i am still in need of a way to calculate it for a non-centroid centre-of-mass, as i can't always guarantee to have things in parts.
 
  • #4
Hi Nanako! :smile:
Nanako said:
hi tim, thanks for the help!

I was aware of the PA theorem, but i had already excluded it as being useful to me, I'm not fully understanding how it relates to my needs here. This theorem seems to largely depend on already knowing I through the object's centre of mass, first.

yes of course …

you have to know I through the part's centre of mass, first :smile:
it mentions 2d planes and the centroid, but it assumes even distribution of mass there, (and therefore, assumes that the centroid is also the centre of mass).

if the distribution isn't even, then those standard formulas won't apply, and you'll have to do a difficult integration anyway :redface:
The latter part might be useful for my two-part hammer idea, although how do i combine I on two parts like the hammer example.

also i am still in need of a way to calculate it for a non-centroid centre-of-mass, as i can't always guarantee to have things in parts.

i'ts very unusual for something not to be made of parts which are of uniform density …

so just split it into such parts, and use the parallel axis theorem (and your table) on each part (and then add all the results)
 
  • #5


Hi there, calculating moment of inertia for 2D rectangles can be a bit tricky, but there are a few ways to approach it. One method is to use the parallel axis theorem, which states that the moment of inertia of an object can be calculated by adding the moment of inertia for an axis through the object's center of mass to the moment of inertia for an axis parallel to the original axis and passing through the center of mass.

In the case of a rectangle, the moment of inertia can be calculated by using the formula I = (1/12) * m * (h^2 + w^2), where m is the mass of the object, h is the height, and w is the width. This formula assumes that the axis of rotation passes through the center of mass.

However, if the axis of rotation does not pass through the center of mass, you can use the parallel axis theorem to calculate the moment of inertia. For example, if you have a rectangle with an arbitrary center of mass, you can divide it into smaller rectangles and calculate the moment of inertia for each one using the formula above. Then, you can use the parallel axis theorem to add these moments of inertia together to get the total moment of inertia for the object.

In the case of the hammer, dividing it into two rectangles with evenly distributed mass may simplify the calculations, but it may not be entirely accurate. It would be better to divide the hammer into smaller rectangles with varying masses and use the parallel axis theorem to calculate the moment of inertia for each one.

As for references, you can check out textbooks or online resources on classical mechanics or rotational dynamics for more information on calculating moment of inertia for 2D objects. I hope this helps and good luck with your physics engine!
 

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to rotational motion. It is calculated based on an object's mass, shape, and distribution of mass.

How is moment of inertia calculated for a 2D rectangle?

The moment of inertia for a 2D rectangle can be calculated using the formula: I = (1/12) * m * (h^2 + w^2), where m is the mass of the rectangle, h is the height, and w is the width.

Do all sides of a rectangle have the same moment of inertia?

No, the moment of inertia will vary depending on the distribution of mass in the rectangle. A rectangle with more mass toward its edges will have a larger moment of inertia than one with more mass toward its center.

How does the moment of inertia change if the rectangle is rotated?

If the rectangle is rotated around its center of mass, the moment of inertia will remain the same. However, if it is rotated around a different axis, the moment of inertia will change and can be calculated using the parallel axis theorem.

What is the significance of moment of inertia in physics and engineering?

Moment of inertia is an important concept in physics and engineering, as it helps in understanding an object's motion and stability. It is used in the design of structures and machinery, and is also important in analyzing rotational motion and calculating angular momentum.

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