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## Main Question or Discussion Point

Hello,

I am having difficulty understanding how the first moment of area is a useful quantity, or even how the formula represents its physical interpretation.

So apparently the physical meaning of the first moment of area is where the center of mass of a uniformly dense object is, and that is found by multiplying an infinitesimal area by the distance of that area from the axis. This somehow is interpreted as a centroid when summed up over the surface. To me, this multiplication doesn't really mean anything, and it has units of volume, so that definitely doesn't give me any physical interpretation of a centroid, which I would think would be some coordinate.

It appears that a force acting further away from the center of the object is greater, which sort of makes sense to me, since the torque is greater further from the axis of revolution, but I think that is a consequence of the lever arm, not the actual force itself acting at a certain point. Why wouldn't the force be uniform? I think of a beam where I have an intuition that the further away from the pole by which it begins would require a greater force to keep up, but I can't explain that intuition.

Since I don't understand the 1st moment of area, surely there is no way for me to understand the second moment of area. Somehow squaring the distance by that infinitesimal area gives a useful quantity called ''inertia''. Is this somehow related to the other use of the word inertia that I remember being ''an object's resistance to a change in motion'' or something akin to that.

If anyone can help elucidate these two concepts, I would be most grateful

I am having difficulty understanding how the first moment of area is a useful quantity, or even how the formula represents its physical interpretation.

So apparently the physical meaning of the first moment of area is where the center of mass of a uniformly dense object is, and that is found by multiplying an infinitesimal area by the distance of that area from the axis. This somehow is interpreted as a centroid when summed up over the surface. To me, this multiplication doesn't really mean anything, and it has units of volume, so that definitely doesn't give me any physical interpretation of a centroid, which I would think would be some coordinate.

It appears that a force acting further away from the center of the object is greater, which sort of makes sense to me, since the torque is greater further from the axis of revolution, but I think that is a consequence of the lever arm, not the actual force itself acting at a certain point. Why wouldn't the force be uniform? I think of a beam where I have an intuition that the further away from the pole by which it begins would require a greater force to keep up, but I can't explain that intuition.

Since I don't understand the 1st moment of area, surely there is no way for me to understand the second moment of area. Somehow squaring the distance by that infinitesimal area gives a useful quantity called ''inertia''. Is this somehow related to the other use of the word inertia that I remember being ''an object's resistance to a change in motion'' or something akin to that.

If anyone can help elucidate these two concepts, I would be most grateful

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