# Is it okay to use the same moment of inertia formula for both a door turning on hinge and a long thin rod rotating at its end?

My book says you can redistribute the mass elements of a object to simplify its moment of inertia formula.

But squeezing a door into a rod would change its density. Does it matter?

Moments of inertia are additive, so if you have lots of separate elements you can just sum up all the individual moments of inertia to get the total.

In this case you can regard your door as being made up of lots of rods:

OK it's a slightly odd looking door, but the point is that if the door is (conceptually at least) made up by stacking $N$ rods then its moment of inertia is just $N$ times the moment of inertia of one rod.

• Or you don't have stack up the rods. Just move move the mass elements parallel to the axis of rotation, the middle part of the door, made from an expanded (clay) rod, has a higher density than the front and back. Nov 29, 2015 at 20:17

When calculating the moment of inertia of a mass along an (ideal) axis, the only thing that matters is how much mass is at any given distance from the axis. The extent and distribution of mass either parallel to the axis or circumferential to the axis is immaterial.

Your example of a door (a rectangular prism) versus a rod is good; assuming the masses and the maximum lengths perpendicular to the axis are the same, and the thicknesses of the door and the rod are small relative to that length, then the moment of inertia will be the same.

Another example would be a ball on a weightless rod, versus a circumferential hoop. Again, same moment of inertia.

• Not maximum distance I think. You can only move they Parallel to the rotation axis. Nov 29, 2015 at 16:15
• What's Small relative? Nov 29, 2015 at 16:15
• @Doeser I updated the text a bit; let me know if it still isn't clear. Nov 29, 2015 at 16:17
• So density doesn't matter? All that matters is distance of all mass elements to the axis? Nov 29, 2015 at 16:19
• Well, why would the density matter? Does it matter for linear inertia, given equal masses? Nov 29, 2015 at 16:20