# 1st, 2nd and mass moment of inertia

• gravity121
In summary, there are various terms and formulas used to describe the distribution of mass and its resistance to different forces and motions. These include the first and second moments of area and mass, which are used to determine the centroid and center of gravity, as well as the resistance to bending and angular acceleration. The terms and formulas can be confusing, but understanding their physical significance can help in their application.
gravity121
I know most of the formula's but I'm trying to get a conceptual understanding of everything and finding it very hard to get concise information about everything all in one place especially when there are so many names for the same thing. Most resources just rehash the formulas without explaining what's going on. Please help. I'm looking for the physical understanding of what's going on and clarity on the formulas. Please fill in and correct any mistakes below or confirm if I'm understanding correctly. Thank you

1st moment of area - used to predict resistance to shear stress

Qx - area distribution around x-axis and resistance to shear stress about x-axis? = int yda
Qy - area distribution around y-axis and resistance to shear stress about x-axis? = int xda
Qz - area distribution around z-axis and resistance to shear stress about z-axis? = ?
is this the polar ist moment of area, is there such a thing? Is this correct??
Is there a product 1st area moment? If so, what is it for? = ?

2nd moment of area - a cross sectional resitance to bending
Ixx - resistance to bending about x-axis due moment about x-axis = int y^2 da
Iyy - resistance to bending about y-axis due to moment about y-axis = int x^2 da
Ixy - resistance to bending about x-axis due to moment about y-axis (assuming moment is not applied on through the centroid or this would be zero) = int yx da
Iyx - as above
Iz - resistance to torsion (polar 2nd moment of area) = = int (x^2 + y^2 da) = Ixx + Iyy

Moment of Inertia
Ixx - resistance to rotation about x-axis due moment about x-axis = int y^2 dm
Iyy - resistance to rotation about y-axis due to moment about y-axis = int x^2 dm
Ixy - resistance to rotation about x-axis due to moment about y-axis (assuming moment is not applied on through the center of gravity or this would be zero) = int xy dm
Iyx - as above
Iz - resistance to resistance to rotation about z-axis due to moment about z-axis. Is this also called the polar moment of inertia. = int (x^2 + y^2) dm ?

Is there any other moments of something out there, like volume or whatever?

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gravity121 said:
I'm looking for the physical understanding of what's going on and clarity on the formulas. Please fill in and correct any mistakes below or confirm if I'm understanding correctly.

I agree, the range of terms is confusing, and my lack of TeX skills is about to make this equally confusing...

How's this- for a planar section 'S' of an object with density 'ρ', the center of mass 'M0' and first moments of area 'A1' and mass 'M1' are:
M0 = ∫ρdS, A1=∫x dS and M1 = ∫xρ dS

You have Q = A1 above.

Second moments involve tensor-valued integrands, but I'll do my best. The inertia tensor, which generates second moments A2 and M2, is:

N = ∫[(xx)I-xx]ρ dS

This will generate your second moments of area (omit the density) and moments of inertia above. I expect you could continue this to a third moment, but they don't mean anything (AFAIK). M0 corresponds to the response of a body to a force F, while A1, M1 and N relate to the response of a body to the 'moment' x×F. This 'moment' is also sometimes called a 'force couple' or 'torque':
http://physicsnet.co.uk/a-level-physics-as-a2/mechanics/moments/

Now, if you consider the stress tensor instead of forces, you have a 'stress couple', which is generally taken to be zero:
http://elibrary.matf.bg.ac.rs/bitst...ty_with_couple_stress_ToupinRA.pdf?sequence=1

Does this help?

It does a little but it doesn't clarify the physical understanding for me, as in what each term represents. What I wrote originally, is it correct or incorrect?

gravity121 said:
It does a little but it doesn't clarify the physical understanding for me, as in what each term represents. What I wrote originally, is it correct or incorrect?

Hows this- if a body's interia tells you how much mass there is, the interia tensor tells you how it is distributed in space. The inertia tensor relates to rotational motion, but also deformation (not just bending).

gravity121 said:
It does a little but it doesn't clarify the physical understanding for me, as in what each term represents. What I wrote originally, is it correct or incorrect?
I wonder whether there has to be a connection except in the common form of the formula.

Thank you but I'd really appreciate if you could confir, or correct each of the assumptions above??

Cheers

In statics, the second moment of area is sometimes called, confusingly, the moment of inertia.
(MOI is technically the second moment of mass.)
Similarly, the first moment of area is sometimes called the moment of mass.

First moment of area -- Used to find centroid of a plane figure, for example
Second moment of area -- A beam cross section's resistance to bending
First moment of mass -- Used to find an object's center of gravity
Second moment of mass -- Resistance to angular acceleration

## 1. What is the definition of first moment of inertia?

The first moment of inertia, also known as the centroidal moment of inertia, is a measurement of an object's resistance to rotational motion around a specific axis. It is calculated by multiplying the mass of each point in the object by its perpendicular distance from the axis of rotation and summing these values.

## 2. What is the significance of second moment of inertia?

The second moment of inertia, also known as the area moment of inertia, is a measurement of an object's resistance to bending or deflection. It takes into account not only the mass of an object, but also its distribution of mass around an axis. It is an important parameter in structural analysis and design.

## 3. How is mass moment of inertia different from second moment of inertia?

Mass moment of inertia is a measurement of an object's resistance to rotational motion due to its mass distribution. It is calculated by integrating the second moment of area over the object's volume. In contrast, second moment of inertia only takes into account the area distribution of an object and does not consider its mass.

## 4. How does the moment of inertia affect an object's stability?

An object with a larger moment of inertia will be more stable and less prone to rotational or bending motion. This is because it requires more force to overcome its resistance to motion. On the other hand, an object with a smaller moment of inertia will be less stable and more easily influenced by external forces.

## 5. How can the moment of inertia be calculated for irregularly shaped objects?

The moment of inertia for irregularly shaped objects can be calculated using calculus and the parallel axis theorem. This involves dividing the object into small elements and calculating the moment of inertia for each element. The total moment of inertia is then found by adding up the individual moments of inertia. Alternatively, computer software can be used to calculate the moment of inertia for complex shapes.

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