First Moment of Area: Definition, Formula, Uses, Calculation

In this article, you will learn a complete overview of the first moment of area such as its definition, formula, units, calculation, and many more. 

The first moment of area, sometimes misnamed as the first moment of inertia, is based on the mathematical construct of moments in metric spaces.

It is a measure of the distribution of the area of a shape in relation to an axis.

I have already discussed in the previous article the second moment of area, the polar moment of inertia, etc.

Here only discussed the first moment of area so without wasting time let's get started.

What is the First Moment of Area?

It is the Summation of area times the perpendicular distance between the centroid and an axis (x or y).

It is denoted by Q.

Mathematically,

Moment of Area,

Q = d × A

Where,

d = Perpendicular Distance

A = Area

The first moment of area about an x-axis,

Qx = Σ(A.ȳ)

Where,

A = Area

ȳ = centroidal distance perpendicular to the x-axis

The first moment of area about a y-axis,

Qᵧ = Σ(A.x̄)

Where,

x̄  = centroidal distance perpendicular to the y axis

A = Area

Units 

In the SI Unit,

As we know, the first moment of area

Q = d × A

So,

Q = m × m²

Q = m³ or mm³

Similarly,

In the CGS unit, 

Q = cm³

In the FPS unit,

Q = ft³ or inch³

So, The first moment of area has the unit of length to the third power.

Uses

There are the following uses of the first moment of area:
  1. To calculate the centroid area
  2. To find the second moment of area

How to Find the First Moment of Area of the Whole Body

Let a small elemental area dA and x̄ and ȳ be the distance to that elemental area measured from a given x and y-axis. 

first moment of area

Now, the first moment of area in the x and y directions are respectively given by:

Q = d × A

So, the first moment of area x-x axis,

Qx = ȳ.dA

The first moment of area y-y axis,

Qᵧ = x̄.dA

Now,

The first moment of area of the whole element will be known by the integration of the whole body.

So, the first moment of area of the whole element in x direction:

Qx = ∫ȳ.dA

Similarly,

The first moment of area of the whole element in y direction:

Qᵧ = ∫x̄.dA

How to Find a Centroid Through the First Moment of Area

Let an area A, of any shape, and division of that area into n number of very small, elemental areas. Let x̄ and ȳ be the centroidal distances to each elemental area measured from a given x-y axis.

So, the total moment of area about a y-axis,

A.x̄ = A1.X1 + A2.X2 + A3.X3 + A4.X4 + .. 

x̄ = (A1.X1 + A2.X2 + A3.X3 + A4.X4 + .. )/A

It can also be written as,

x̄ = Σ(A.x)/A

Similarly, So, the total moment of area about an x-axis,

ȳ = Σ(A.y)/A

First Moment of Area for Different Section

Circle and Hollow Circle

As we know,

Q = d × A

first moment of area for circle and hollow circle

Area of a circle,

π r²

So, 

The first moment of area for the circle will be,

Q = π r² × d/2

Where,

d/2 = centroidal distance of the circle.

Similarly for hollow circle

Q = π (R²- r²) × D/2

Where,

D/2 = centroidal distance of the circle.

Rectangular and Hollow Rectangular

As we know,

Q = d × A

first moment of area of rectangle and hollow rectangle

Area of Rectangle,

b.d

So,

Q = b.d × d/2

Where,

d/2 = centroidal distance of the rectangle.

Similarly hollow rectangles,

Q = (B.D - b.d) × D/2

Where,

D/2 = centroidal distance of the circle.

Calculation

Question

Calculate the first moment of area about the x-axis as shown in the figure.

calculation of first moment of area

Solution

From figure,

Component A,

A = 400

ȳ = 55

Aȳ = 22000

Component B,

A = 400

ȳ = 30

Aȳ = 12000

Component C,

A = 300

ȳ = 5

Aȳ = 1500

So,

As we know, the first moment of area is about an x-axis.

Qx = Σ(A.ȳ)

Qx = 22000 + 12000 + 1500

Qx = 35500 mm³


So here you have to know all aspects related to the first moment of area

If you have any doubts then you are free to ask me by mail or on the contact us page.

Thank You.

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