Why Use Area Instead of Mass in Moment of Inertia Formula?

[yecko]

Homework Statement

Homework Equations

I=1/12*bh^2+Md^2

The Attempt at a Solution

for the highlighted 600mm^2 (and the 300mm^2 below) in the middle of the page,
when finding the moment of inertia, what the solution used is an area, yet for related formula required the use as for mass in the part Md^2.
why should area is used here? is Ad^2 having a contradict unit with moment of inertia?
sorry for unable to type out the whole problem statement out and thank you very much

 

[mjc123]

If this was a thin slice with this profile, you could find the MI using the mass, with mass = area* thickness* density. If the density is uniform, the MI is proportional to the area, so you don't need to know the thickness and density - you can work it out using the area, and the results are (geometrically) the same. I in this case is called (I think) the second moment of area, and is what you use in cases like this, where the stress depends on the geometry of the beam but not its mass. (You would need MI if you were looking at e.g. rotational kinetic energy).

 

[Dr.D]

The bending moment equation is
bending stress = moment * distances to most distant fibre/(area moment of inertia)
There is no need to discuss thin slices, etc. Simply calculate the area moment of inertia with with respect to the neutral axis and move on.

 

[David Lewis]

Moment of inertia is a misnomer borrowed from dynamics. In statics it should be the second moment of area. If you have a rotating mass, angular inertia = ∑ dm x². But the second moment of area is ∑ dA x². The formulae have a similar format.

 

[Dr.D]

Agreed that the term "second moment of area" is correct, but to call "area moment of inertia" a misnomer is a bit of a stretch. Both usages are long established, and not really confusing to anyone who pays attention. I've heard both of these terms all of my professional life, but then, I'm new to the area, only 50 year since my PhD in mechanics.

 

[haruspex]

Agreed that the term "second moment of area" is correct, but to call "area moment of inertia" a misnomer is a bit of a stretch. Both usages are long established, and not really confusing to anyone who pays attention. I've heard both of these terms all of my professional life, but then, I'm new to the area, only 50 year since my PhD in mechanics.

The OP did not write "area moment of inertia":

when finding the moment of inertia

Thus, it was strictly incorrect, and that inexactitude strongly relates to the OP's confusion.

Beyond that, yes it is and always has been a misnomer to describe it as some kind of inertia, for obvious reasons. We should not perpetuate such barriers to understanding just because we old guard are used to it.

 

[Dr.D]

Beyond that, yes it is and always has been a misnomer to describe it as some kind of inertia, for obvious reasons. We should not perpetuate such barriers to understanding just because we old guard are used to it.

Oh, my! How could I possibly challenge such magisterial authority! Mea culpa, mea culpa, mea maxima culpa!

 

[haruspex]

Oh, my! How could I possibly challenge such magisterial authority! Mea culpa, mea culpa, mea maxima culpa!

Don't forget the Hail Marys.