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This is something I have never understood despite applying the formulas in many classes. What I'm hoping to find here is some clarification.

My understanding is that the first moment of inertia is torque, and the second is an object's resistance to rotation based on how the mass is spread out.

Then, the moments of area have to do with geometrical spread and thus aren't necessarily related to moments of inertia, but they typically go hand in hand when material density is constant.

But I'm not confident that these definitions are correct and I don't understand the difference between 1st and 2nd moments of area or where the centroid comes into play or whether there are more moments; 3rd, 4th.

What I am asking for is clarification and a conceptual definition about what each is and how they are related.

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Moments of area is defined by the order of the radial distance being taken for the Moment calculation;

From engineeringpsycho.blogspot.com:

(a)The first moment of area equals the summation of area times distance to an axis [Σ(a x r)]. It is a measure of the distribution of the area of a shape in relationship to an axis. First moment of area is commonly used in engineering applications to determine the centroid of an object or the statical moment of area.

For example if you have a circular disc and take the first moment about an axis passing through its centre the you should get the first moment to be zero i.e. in statical condition your disc can be static on a pin-head placed at the centre.(a definition of centroid)

From engineeringpsycho.blogspot.com:

b) The second moment of area, also known as the area moment of inertia or second moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection which are directly proportional.

second moment of area= Ar^2

Mass being a measure of Inertia -whenever the masses /distribution of mass about an axis is taken for moment calculation - one designates the term as "moment of Inertia" and it has important role -very much analogous to the behaviour of mass

From engineeringpsycho.blogspot.com:

c) Mass moment of inertia [...] is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation.

Mass Moment of inetia = mr^2

Higher order moments are used in analysing the motion say in bodies of definite shapes or in turbullent flow of liquids.... see for higher order moment of Inertia see "The effect of third-and fourth-order moments of inertia on the motion of a solid"

From the abstract:

The problem of the effects of higher-order moments of inertia on the motion of a solid, fixed at the centre of mass and having a spherical central ellipsoid of inertia in a central Newtonian field of force is investigated.

Uniform bodies of the simplest geometrical shapes (a cube, cone and cylinder) are considered. In view of the difference in the symmetries of these bodies the nature of their motions will be different. The equations of motion of a cone and a cylinder are integrated in terms of ultra-elliptic and hyperelliptic functions respectively. Sets of positions [...]

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