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 ?
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.
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 $\vec{r} \times \vec{F}$. As @Apurba said, $\sum{\vec{F}}$ may not be zero.
Moment = Magnitude of Force x Perpendicular distance to the pivot.
Torque and moment are essentially the same thing and are calculated in the same way - it's really the context that determines which word is used. 'Torque' is usually used when we're talking about the twisting effect on a shaft and 'moment' is usually used when we're talking about the bending effect on a beam. If you're using a spanner to tighten a bolt, we would say that your hand exerts a moment on the end of the spanner but the spanner exerts a torque on the head of the bolt.
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.
Moment is the more general term which means quantity evaluated when something is multiplied by its moment arm (perpendicular distance).
Some examples of moments:
So is torque equivalent to moment of force? In my opinion no, because the above moments require a generating vector (force, rotation, impulse, and momentum) to be present. But you can have torque without a force, but with a force couple. I prefer to use the term pure torque instead of force couple because in this case a torque vector $\vec{\tau}$ can stand on its own, without needing to define the details of the force couple (force, separation, and direction).
So torque can have one of two meanings depending on the context
$$ \text{(torque)} = \begin{cases} \vec{r}\times \vec{F} & \text{(moment of force)} \\ \vec{\tau} & \text{(pure torque)} \end{cases} $$
For example, a shaft carries a pure torque, but a lever transfers a force moment from one end to the other.
moment is turning effect produced by a force . while torque is due to rotation of body.
Moment is bending due to linear force and the distance from the axis is perpendicular whereas in torque rotation takes place beyond 360 degrees.
