What is the difference between first and second moments in mechanics?

[chetzread]

Homework Statement

in my book , it stated that first moment is moment of mass , second moment as moment of inertia , is it correct ?

Homework Equations

The Attempt at a Solution

IMO , first moment can also be called as first moment of inertia , with formula of Ix or Iy , whereas for second moment of inertia , it has formula of Ixx or Iyy.

 

[chetzread]

I have an example here , the question is find the moment inertia .
I am not sure whether it is either first or second moment on inertia , how to determine it ? Btw , the author considered it as second moment of inertia ? why not first moment of inertia ?

 

[haruspex]

I feel the book's definitions are wrong. The terms mass and inertia are, or should be, interchangeable.
You can find references online to first and second moments of mass.
That said, unqualified "Moment of Inertia" refers always to the second moment.

 

[chetzread]

I feel the book's definitions are wrong. The terms mass and inertia are, or should be, interchangeable.
You can find references online to first and second moments of mass.
That said, unqualified "Moment of Inertia" refers always to the second moment.

so, the example in book i wrong? by saying moment of inertia, we have no enough evidence to show that whether it's first or second moment ?

 

[haruspex]

so, the example in book i wrong? by saying moment of inertia, we have no enough evidence to show that whether it's first or second moment ?

No I wrote that "Moment of Inertia" should be taken to mean the second moment unless some other order of moment is specified.

 

[David Lewis]

in my book, it stated that first moment is moment of mass, second moment as moment of inertia, is it correct ?

Yes, it's correct but poorly explained.

First moment means the distance from the reference axis to each element of area or mass is raised to the first power.
Second moment means the distance to each element of area or mass is raised to the second power.

So you see no matter what kind of moment you have (first or second), it may either be a moment of area, or a moment of mass:

Σ r * dA = first moment of area
Σ r * dm = first moment of mass
Σ r² * dA = second moment of area
Σ r² * dm = second moment of mass = moment of inertia

The problem asks you to find the polar mass moment of inertia for the given body.
You can think of moment of inertia as the resistance to angular acceleration.

 

[haruspex]

Yes, it's correct

As I read it, the book is saying that "moment of mass" means first moment, while "moment of inertia" means second moment. I am sure that is wrong, and it does not accord with the rest of your own post.
Of course, it could be that the book is being misquoted. @chetzread, please post the exact wording.

 

[chetzread]

Yes, it's correct but poorly explained.

First moment means the distance from the reference axis to each element of area or mass is raised to the first power.
Second moment means the distance to each element of area or mass is raised to the second power.

So you see no matter what kind of moment you have (first or second), it may either be a moment of area, or a moment of mass:

Σ r * dA = first moment of area
Σ r * dm = first moment of mass
Σ r² * dA = second moment of area
Σ r² * dm = second moment of mass = moment of inertia

The problem asks you to find the polar mass moment of inertia for the given body.
You can think of moment of inertia as the resistance to angular acceleration.

why it can't be first moment for the case in the attachment i have uploaded earlier ?

 

[haruspex]

why it can't be first moment for the case in the attachment i have uploaded earlier ?

As I posted, if the order of the moment is not specified (zeroth, first, second, ...) then moment of inertia should be taken to mean the second moment.
Where I disagree with the textbook extract you posted is that I do not believe that "moment of mass" defaults to meaning first moment.

 

[chetzread]

As I posted, if the order of the moment is not specified (zeroth, first, second, ...) then moment of inertia should be taken to mean the second moment.
Where I disagree with the textbook extract you posted is that I do not believe that "moment of mass" defaults to meaning first moment.

do you agree that second moment = moment of inertia by default ?

 

[haruspex]

do you agree that second moment = moment of inertia by default ?

Yes, the default meaning of "moment of inertia" is the second moment.

 

[David Lewis]

Moment of mass is understood to mean first moment of mass because (in this context) mass refers to the physical quantity that reacts to the pull of gravity, not resistance to acceleration.

First moment of mass is useful in center of gravity calculations, and can be thought of as the torque that gravity imposes on a static body.

Moment of inertia is taken to mean the second moment of mass (or, in bending analysis, the second moment of area) because here the term mass refers to the quantity that resists acceleration.

 

[haruspex]

Moment of mass is understood to mean first moment of mass

I have not been able to find any reference for that. Can you post a link?

Edit: ok, I finally managed to find a couple of obscure references for this usage.
http://www2.palomar.edu/users/cchamberlin/Math 205 PDF/lecture 16_5.pdf
http://mathwiki.ucdavis.edu/Core/Ca...ultiple_Integrals/Moments_and_Centers_of_Mass
But most discussions of moments do not mention it. Personally, I find the stated reason for discriminating between mass and inertia unconvincing. Centre of mass is not the same, in general, as centre of gravity.