Moment

The moment (or torque) of a force is its resulting rotational tendency, a vector quantity measured in $N\cdot m$. In 2-D, the moment points in the $\hat{k}$ direction, but in 3D points in different directions.

Where the vector $\vec{r}$ is the perpendicular distance between the force and some point $O$:

$${\vec{M}}_O=\vec{r}\times \vec{F}$$ $$\sum M_O=\sum Fd={\dot{H}}_O$$

A key principle in statics analysis is Varignon’s theorem, or the principle of moments, that suggests that the moment about a point is equal to the moments of the components of the force about the point. What this means is that we can decompose and solve for the moment according to $x$, $y$, and $z$ components individually.

In dynamics analysis, we have a handy formula that we can link with relative motion problems:

$$\sum {\vec{M}}_G=I_G\vec{\alpha }$$

where $I_G$ is the mass moment of inertia and $\vec{\alpha }$ (alternatively $\ddot{\theta }$) is the angular acceleration.