Moment of a Force about a Point | Civil Engineering Optional Notes for UPSC PDF Download

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 F₁, F₂, etc. that form a couple, with position vectors (about some origin P) r₁, r₂, etc., respectively. The moment about P is

M = r₁ * F₁ + r₂* F₂ + r₃* F₃ + ..

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

M’ = (r₁ + r)x F₁ + (r₂ + r)x F₂ + (r₃ + r)x F₃ + ...

Now the distributive property of the cross product implies

M’ = (r₁ x F₁ + r₂ x F₂ + r₃ x F₃+ ….) + r x (F₁ + F₂ +F₃ + ….)

However, the definition of a force couple means that

F₁ + F₂ +F₃ +...= 0

Therefore,

M’ = r₁ x F₁ + r₂ x F₂ + r₃ x F₃+ .. =M

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