# First moment of area and second moment of area

• harryiverson
In summary, the first moment of area is used to find the centroid, while the second moment of area (inertia) is used to find the resistance in the x-y plane to torque in the z plane. The polar moment of inertia is different from the second moment of area, as it is defined about the z-axis and is used in cylindrical polar coordinates. The second moment of area is often used in beam bending equations. While there are similarities between the two concepts, they are distinguished by their definitions. The proofs for these concepts can be found online.
harryiverson
i am perplexed as to the first moment of area and second of area; i would like to know
1. why they come (how they are figured out and distinguished from each other)
2. what is meaning of these 2 moment of area in terms of physics

what i have learned is that the first moment of area is used to find centroid and the second Of area (Inertia) is to find the resistance in x-y plane to torque in z plane.

btw i am also confused as to the idea of polar moment of inertia ; how it is different from the second moment of the inertia.

hope there are answers to my stupid questions. i just kind of feel lost.

i have viewed some utube video but they can't totally clear my doubt

How and how much did you learn about those three concepts?

a few line in my structural mechanics lecture notes that state their definition as well as their formula and some example of how they use.

harryiverson said:
what i have learned is that the first moment of area is used to find centroid
that is correct

harryiverson said:
and the second Of area (Inertia) is to find the resistance in x-y plane to torque in z plane.
btw i am also confused as to the idea of polar moment of inertia ; how it is different from the second moment of the inertia.

Be careful, the moment of inertia is NOT the same as the second moment of area. There is often some confusion because they are both denoted by $I$ and both have similar looking integrals.

The moment of inertia is defined as $I = \iint r^2 dm$ and you are correct in saying that we use this when we are thinking about torques applied to a system (analogous to Newton's 2nd law for linear acceleration) - we use $M = I \ddot \theta$. The moment of inertia is always defined relative to a given axis (eg. through the centroid, through the edge, etc). The polar moment of inertia is the moment of inertia about the polar (z) axis ('out of the page axis'). If you are familiar with cylindrical polar coordinates, it is the z-axis.

The second moment of area (also defined relative to a given axis) is defined as $I = \iint r^2 dA$ and this is often used when we are talking about beam bending. I am not sure how familiar you are with beam bending equations, but it is used in equations relating bending moments to shear stress or deflections.

There are some similarities between them (e.g. parallel and perpendicular axis theorems operate in the same way operate for both)

So to sum up:
How are they different? By definition
Where do they come from? There are proofs on the internet that you can search up for these, so no point typing them up here

Lnewqban

## 1. What is the difference between first moment of area and second moment of area?

The first moment of area, also known as the centroid, is a measure of the distribution of an object's area around a specified axis. It is calculated by multiplying the area of each section of the object by its distance from the axis and summing these values. The second moment of area, also known as the moment of inertia, is a measure of an object's resistance to bending around a specified axis. It is calculated by multiplying the area of each section of the object by the square of its distance from the axis and summing these values.

## 2. How is the first moment of area used in engineering?

The first moment of area is used in engineering to determine the centroid of an object, which is important for understanding the stability and balance of structures. It is also used in calculating the shear stress and bending stress in beams.

## 3. What is the formula for calculating the first moment of area?

The formula for calculating the first moment of area is:

M = ∫y*dA

Where M is the first moment of area, y is the distance from the axis to the element of area dA, and the integral is taken over the entire area of the object.

## 4. What factors affect the second moment of area?

The second moment of area is affected by the shape, size, and distribution of an object's cross-sectional area. Objects with a larger cross-sectional area and a greater distance from the axis will have a higher second moment of area, indicating a greater resistance to bending.

## 5. How is the second moment of area used in structural analysis?

The second moment of area is used in structural analysis to calculate the deflection and bending stresses in beams and other structural elements. It is also used in the design of beams and columns to ensure they can withstand the expected loads and forces.

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