Static Equilibrium Conditions

[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...

 

[rezeew]

I would respond by saying that the statement is correct and is based on the fundamental principles of static equilibrium. In static equilibrium, all forces acting on an object must balance out, meaning that the resultant force and the resultant moment (or couple) must both be equal to zero. This is because an object in static equilibrium is not moving or rotating, so the net force and net moment acting on it must be zero.

In this case, the statement is specifically referring to the resultant moment, which is the combination of all the moments (or couples) acting on the object. The position of the point O is arbitrary, but the magnitude and direction of the moment depend on the chosen point. However, since the object is in static equilibrium, the resultant moment must be zero, regardless of the chosen point. This is a fundamental principle of static equilibrium and can be proven mathematically.

To determine if an object is in static equilibrium, we can calculate the resultant moment about any point and if it is zero, then we can conclude that the object is in static equilibrium. This is because if the resultant moment is zero at one point, it must also be zero at any other point due to the fundamental principles of static equilibrium.

In summary, the statement is based on the fundamental principles of static equilibrium and can be proven mathematically. It is a useful concept in determining if an object is in static equilibrium.