# Static Equilibrium Conditions

## 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.

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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'

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.

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