Newton's first law states that an object that remains in uniform motion will remain in uniform motion unless it is acted upon by an external force. This also includes that an object at rest will remain at rest unless it is acted upon by an external force.
When more than one force acts upon an object, the vector sum of these forces is the resultant force.
When the resultant force on an object is zero, it will remain at rest if it is at rest, or continue to move in a straight line at a constant velocity if it is in motion. There is no change in either the magnitude or direction of its velocity. That is, there is zero acceleration.
The above concept can also be applied to motion in any selected direction.
Consider an object moving along the x-axis. If no net force is applied to the object along the x-axis, it will continue to move along the x-axis at a constant velocity with no acceleration.
We can extend this to the y- and z- axes.
In any system, unless the applied forces cancel each other out, that is, the resultant force is zero, there will be acceleration in the direction of the resultant force.
In static systems, where motion does not occur, the sum of the forces in all directions must always equal zero (otherwise, it's a dynamics problem). This concept can be represented mathematically with the following equations.
The concept also applies to rotational motion.
If the resultant moment about an axis is zero, the object will have no rotational acceleration about the axis. If the object is not spinning, it will not start to spin. If the object is spinning, it will continue to spin at the same constant angular velocity. Again, we can extend this to moments about the y-axis and the z-axis. This is represented mathematically with the following.
What happens if the sum of forces or moments is not zero?Edit
If the object has a net moment not equal to zero, it will spin about whichever axes are not zero. If the object is acted upon by a net force not equal to zero, it will accelerate in whatever direction the net force is. Once the object moves it cannot be analyzed with statics, instead the rules of dynamics take control. The acceleration of the object is governed by Newtons second law,
You will never need this equation in Statics.