Geometrical moment of area (First moment of area)

What is the geometrical moment of area?

​

- Area: Visible cross section of the structure

          (Two-dimensional plane)

​

- Moment: Weight by distance of cross-sectional area

             'Area x distance'

 

What is it used for and why is it needed?

Mechanical engineers need to find an intermediate axis for Geometrical shape.

When other factors are constant & only the shape is different,

Geometrical moment of area is used

to find the intermediate axis of the structure (Center of area).

 

Fig.1

 

The middle axis of the plane is called the Centroid.

 

 

What is the standard in the middle axis?

The information given in two dimensions is not a density or any physical quantity.

The area as shown is all of the information given.

Therefore, if there is an arbitrary one-dimensional axis that crosses the section

and the two sections have the same area when the two sections are bisected,

the one-dimensional axis can be said to be the Centroid. (center of the area).

 

Fig.2

 

However, the centroid is not determined just because the area is the same.

See Fig.3 below~

 

Fig.3

 

There are 4 blocks of the same area, but the red dotted line cannot be the city center.

The distance from the centroid acts as a weight.

 

Also, the number of centroid is infinite as the centroid change

as shown in Fig.4 according to the axis setting.

 

Fig.4

 

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So, let's see how to get the centroid.

 

1. Uniform density along the Z-axis

2. and The assumption that the cross-sectional area is

   uniform along the z axis takes precedence.

 

because,
When we study centroid in books, we only worry about aspects
the existence of only Areas is ideal and not real.
The volume is real.

 

When there is a cross section of an object as shown in Fig.5 below

that satisfies the above assumption,

 

Fig.5

 

The method of finding the city center is shown in Eq.1 below.

 

Eq.1 Centroid

 

 

In other words, what is centroid?

Each area is multiplied by the distance from the reference axis,
weighted, summed, and then divided by the total area.

 

At this time, in Equation 1, the sum of the products of each area and distance,

Q, is called the Geometrical moment of area (First moment of area).

 

Eq.2 Geometrical moment of area

 

I'll finish it at this point.

Thank you~~^^