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In mechanical engineering, the torque due to a couple is given by $\tau = P\times d$, where $\tau$ is the resulting couple, $P~$ is one of the force vectors in the couple and $d$ is the arm of the couple. A couple is made up of two forces of the same magnitude. On the other hand, a moment is also given by $M = P\times d$. However, there is only one force involved! How can the resulting torque be of the same magnitude if in one case two, in the other case only one force is involved? |
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Ingo, when you consider the couple, you may put one of the "spouses" at the origin, so his torque is $P\times d_0$ for $d_0=0$, so his torque vanishes. Meanwhile, she is located at a nonzero $d$ so her contribution is $P\times d$ and nonzero. Because his torque is zero, it doesn't matter whether you add him or not. The only difference between the whole couple (including the husband at the origin) and the separate wife at nonzero $d$ is that the total force (not moment) of the whole couple is zero, while the wife separately acts both by torque and by a nonzero ordinary force as well. |
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The reason for the distinction is that a couple has no net force in some direction. If you had a rod floating in space and applied a couple, it would spin in place. This is because you applied the same force in opposite directions, so the overall force in any direction is 0. However, since you didn't apply the force directly in the center, you impart some spin on it If you applied a moment to a floating rod, it would start spinning because you didn't apply the force in its center, but also move off in some direction, because you applied an unbalanced force, and there was some net force in the direction you applied it. |
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Moment and couple are quite similar terms; but whereas, when we consider two equal but unlike forces acting on a body, that is coupled, whereas moment involves only one force. |
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