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I am a beginner in mechanics, and when I was introduced to moments there was a theorem that says "if a set of forces acting on a rigid body has a resultant, then the algebraic sum of these forces about a certain point is equal to the moment of the resultant about this point" so what I understood is if the resultant of forces is zero then the moment of these forces should be zero too by applying the previous theorem and also because M=rxR so if R equals zero then the moment equals zero too, but then I learned couples, in couples the resultant equals zero but the moment doesn't equal zero how is this if M=rxR then moment should be zero. I hope that you understand my question and I will be thankful for anybody who answers it.

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    $\begingroup$ Look up "couple of forces". $\endgroup$
    – nasu
    Nov 16 '20 at 20:41
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To prove your first statement when the resultant is non-zero, you use the fact that the resultant force acts along a unique line in space. In fact, that line is where the moments of the individual forces sum to zero.

If the resultant force is zero, the whole calculation doesn't make any sense because there isn't a "unique line in space" that you can choose.

As a simple example in two dimensions, suppose you have a unit force acting in the +X direction at point (X = 0, Y = 1) and a unit force in the -X direction at point (X = 0, Y = -1). It should be obvious that the combined moment of the two forces about the origin (X = 0, Y = 0) has magnitude 2, and in fact it also has magnitude 2 about any other point in the plane.

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  • $\begingroup$ Do you mean if resultant of forces equals zero then this theorem doesn't apply because there is no line of action of resultant, and thank you for your answer $\endgroup$
    – Eman
    Nov 16 '20 at 21:08
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The moment of the resultant about a given axis is not equal to the sum of the moments of the individual forces. For this to be true, the moment arm of each of the forces would have to be equal to the moment arm of the resultant (which typically won't be the case). For that matter, what does the moment arm of the resultant even mean?

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