I was watching this lecture on analysis of stress for mechanics of materials. At time 7:20, the lecturer says that in equilibrium, the sum of forces and "moments" in each direction (x,y,z) must be zero. What exactly is meant by "moments" in this context? Does this mean the "twisting" forces? If so, why are twisting forces counted separately from the other forces in the x,y, and z directions?
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Some engineering texts use "moment" and "couple" to talk about forces that tend to rotate an assembly (what physicist mean when they say "torque", but the engineers sometimes have a slightly different meaning for that word).
A roughly translation guide is...
A "couple" is a pair of opposite forces whose points of action are not co-linear. A couple is sometimes called a "pure torque" because it imparts a tendency to rotate without imparting a tendency to accelerate, and engineers will occasionally shorten this to just "torque", which is why a physicist needs to be careful in talking about these things with engineers.
A "moment" is the tendency to rotate imparted by a off-center force (i.e. it is a "torque" in physicist-speak), but because it has not been paired off in a couple you know that you may also have to worry about the linear acceleration that is implied.
In your case the speaker is just saying that the static conditions,
$$ \sum \vec{F} = 0 $$ $$ \sum \vec{\tau} = 0 $$
apply.
