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**Moment of Force about a Point**

The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P"), and in general when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P: Any point will give the same moment. In other words, a torque vector, unlike any other moment vector, is a "free vector".

(This fact is called Varignon's Second Moment Theorem.)

The proof of this claim is as follows: Suppose there are a set of force vectors **F**_{1}, **F**_{2}, etc. that form a couple, with position vectors (about some origin P) **r**_{1}, **r**_{2}, etc., respectively. The moment about P is

M = r_{1 }* F_{1 }+ r_{2}* F_{2 }+ r_{3}* F_{3 }+ ..

Now we pick a new reference point P' that differs from P by the vector **r**. The new moment is

Mâ€™ = (r_{1} + r)x F_{1 }+ (r_{2} + r)x F_{2 }+ (r_{3} + r)x F_{3 }+ ...

Now the distributive property of the cross product implies

Mâ€™ = (r_{1 }x F_{1 }+ r_{2 }x F_{2 }+ r_{3 }x F_{3}+ â€¦.) + r x (F_{1 }+ F_{2 }+F_{3} + â€¦.)

However, the definition of a force couple means that

F_{1 }+ F_{2 }+F_{3} +...= 0

Therefore,

Mâ€™ = r_{1 }x F_{1 }+ r_{2 }x F_{2 }+ r_{3 }x F_{3}+ .. =M

This proves that the moment is independent of reference point, which is proof that a couple is a free vector.

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