Force moment is defined as $\vec{M}=\vec{r}\times\vec{F}$ when force is perpendicular to the lever arm.
How do we know that it has to be this way? How do we know it is not $F\cdot r^2$ etc. ?
My reasoning is sth like things don't rotate when the force is applied through the CM/axis of rotation of an object. So the new physical quantity has to know to be zero then. However it doesn't explain why multiply by $r^1$, $r^3$ is gonna work well in this case.
I might be looking for a historical explanation, don't know precisely what I am looking for. Something bugs me there. Happy to hear your thoughts in general!
