# First moment of area and second moment of area

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.

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

Homework Helper
Gold Member
How and how much did you learn about those three concepts?

harryiverson
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.

Master1022
what i have learned is that the first moment of area is used to find centroid
that is correct

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