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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 ?

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/16389/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Feb 19, 2013 at 13:16
  • $\begingroup$ I upvoted all answers. Since, I am getting different answers I accepted the one that seems most reasonable to me. $\endgroup$
    – Amr
    Commented Feb 19, 2013 at 18:53

7 Answers 7

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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.

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  • $\begingroup$ Thank you for referring to the big picture. The terminology looks alright for me. As far as I understood, this is just a redundancy in terms. I have been told before that torque is different from moment of a force. Is this true ? $\endgroup$
    – Amr
    Commented Feb 19, 2013 at 17:50
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    $\begingroup$ There might be some slight differences, but they probably stem from technical jargon (so no real physical difference). From what I've read (on this website among others), the term "torque" is usually preferred when speaking of the moment of a couple of forces (so when 'twisting' rather than 'rotating'). The term "moment" is used in any other general case. Personally I think it is an unnecessary distinction and source of confusion. I'm not a native English speaker and in my language we don't have this problem. :) $\endgroup$
    – Wouter
    Commented Feb 19, 2013 at 18:24
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    $\begingroup$ In addition, what is the moment of rotational velocity? It is linear velocity $\vec{v} = \vec{r}\times\vec{\omega}$. In fact, both forces and rotations act along a line, the position of which is given by $$ \vec{r} = \frac{\vec{v}\times\vec{\omega}}{|\vec{\omega}|^2} \\ \vec{r} = \frac{\vec{\tau}\times\vec{F}}{|\vec{F}|^2}$$ Do you see the similarity? $\endgroup$
    – John Alexiou
    Commented Nov 27, 2013 at 18:54
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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.

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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.

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  • $\begingroup$ Thus, every moment is a torque $\endgroup$
    – Amr
    Commented Feb 19, 2013 at 11:22
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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.

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Moment is the more general term which means quantity evaluated when something is multiplied by its moment arm (perpendicular distance).

Some examples of moments:

  • Moment of force (torque): $\vec{r} \times \vec{F}$
  • Moment of rotation (velocity): $\vec{r} \times \vec{\omega}$
  • Moment of impulse: $\vec{r} \times \vec{J}$
  • Moment of momentum (angular momentum): $\vec{r} \times \vec{p}$

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.

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

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    $\begingroup$ Neither of these brief definitions provides enough detail to be useful in any way. $\endgroup$
    – Brandon Enright
    Commented May 16, 2013 at 7:05
  • $\begingroup$ and what's the difference between "turning" and "rotation" ? $\endgroup$
    – Amr
    Commented Nov 8, 2013 at 18:40
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Moment is bending due to linear force and the distance from the axis is perpendicular whereas in torque rotation takes place beyond 360 degrees.

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