Difference between torque and moment

Question

What is the difference between torque and moment? I would like to see mathematical definitions for both quantities.

I also do not prefer definitions like "It is the tendancy..../It is a measure of ...."

To make my question clearer:

Let $D\subseteq\mathbb{R}^3$ be the volume occupied by a certain rigid body. If there are forces $F_1,F_2,....,F_n$ acting at position vectors $r_1,r_2,...,r_n$. Can you use these to define torque and moment ?

Answer 1

The moment of a vectorfield $\vec{v}$ at a position $\vec{r}$ is equal to $$\vec{r}\times\vec{v}.$$ So torque is simply a special case where the vectorfield we look at is the force field, $\vec{v} = \vec{F}$. Another way of saying this is that torque is the moment of force.

Answer 2

While the formulas are similar, Torque relates to the axis of rotation driving the rotation, while moment relates to being driven by external force(s) to cause the rotation. Moment is a general term and when used in context of rotational motion is pretty much the same.
Torque is $\vec{F} \times \vec{r}$. As @Apurba said, $\sum{\vec{F}}$ may not be zero. Moment = Magnitude of Force x Perpendicular distance to the pivot.

Answer 3

Torque is $\vec{F} \times \vec{r}$ but in this case $\sum{\vec{F}}$ may not be equal to zero. Where as in case of moment the two equal force acts in tow different side, So $\sum{\vec{F}} = 0$. I think this is the difference.

Answer 4

moment is turning effect produced by a force . while torque is due to rotation of body.