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Equations Of Static Equilibrium


Basic Principle


If we have an object which is stationary and not accelerating, then it does not have any net force applied to it, we know this because of newtons first law of motion. If there is no net force applied to the object, then all of the external forces on this object must cancel each other out. For example, you could have a 2 forces pushing the object in opposite directions with equal magnitudes. The same thing can be said about moments or torques on the object. They all must be equal to 0 if the object is not accelerating.

Sum Of Forces


The summation of forces on a static object is equal to 0, therefore we have the following equation:
[math]\Sigma F = 0[/math]

It is a very simple equation but very useful, the forces plugged into this equation should be in vector form. You can break this equation down into three separate equations one for each direction (x, y and z) so we now have three equations
[math]\Sigma F_x = 0[/math], [math]\Sigma F_y = 0[/math] and [math]\Sigma F_z = 0[/math]

Each one of these equations is true if the object is not accelerating. It is often easier to break the forces into their x, y and z components and use these three equations because you work with less numbers at once.

Sum Of Moments


The sum of moments about a point is very similar to the previously discussed sum of forces. we have the equation:
[math]\Sigma M_P = 0[/math]

where we take the sum of all the moments around a point P which can be any point on the object. It is often easiest to choose a point which has the most unknowns forces acting at that point because a moment is defined as [math]M = Fd[/math] where d is the distance, if the distance is 0, the moment applied at that point due to the force is also 0. you can also break this one down to three separate equations: sum of moments around the x, y or z axis equals 0.