Why Is Moment Zero at Any Point if Zero at One Point in Static Equilibrium?

[emohabatzadeh]

Homework Statement

we know that, every force systems can be generally replaced by a resultant force(R) and a couple(M) at a point O and the position of point O is optional.
but magnitude and direction of M is dependent to this point while magnitude and direction of R is independent.
In static equilibrium R and M are zero at an optional point O. now this is the question:
While M is zero at an optional point O, why should we conclude that M would be zero at every point chosen( infinite in number of points)...please pay attention that " magnitude and direction of M is dependent to the point chosen"...
I mean we don't know the object is in static equilibrium or not and we want to determine it... why do we consider that if M is zero about a point, it means that it is zero about any point? is there a theorem about this? is it provable?

Homework Equations

Static equilibrium conditions.

The Attempt at a Solution

In fact I have no answer to the question...it's not a numerical problem.

 

[kushan]

Let us suppose net torque (couple M)

M = r₁ X F₁ +r₂ X F₂ ...
where ref. point is O

Take another ref point O'
then

M'=(r₁ + r) X F₁ +(r₂ + r )X F₂ ...
Simplifying

M' = r₁ X F₁ +r₂ X F₂ ... + r X ( F₁ + F₂ ...)
However F=F₁ + F₂ ... =0
Hence M=M'

 

[voko]

The above can be compactly phrased this way: if R = 0, then M is independent of O. So R = 0 and M = 0 about any O imply equilibrium.

 

[emohabatzadeh]

Thank You all...that's right...