Calculating Moment of Inertia (2D rectangles)

[Nanako]

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.

 

[tiny-tim]

Hi Nanako!

Use http://en.wikipedia.org/wiki/Parallel_axis_theorem

 

[Nanako]

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.

 

[tiny-tim]

Hi Nanako!

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

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

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)