Statics: 3D Resultants

 Let's state a door problem and try to figure out what the moment is for the problem.

So I'm going to figure this out real quick. There's a door hinge at the negative x direction. and a force pulling 500 newtons at point A 750mm up and 850mm in the negative x direction, with a 20 degree angle between the door and the xy plane. The board is 800mm long.

So we want the moment about the OC axis, which is equivalent to the total moment dot product with the vector of OC. The total moment is the Force of A which is the difference between A to B (B - A) dot product with the vector from O to the spot of the force, which is. The force is 500N in total so we want to figure out the proportion which is (-850, -752, 476)/((850 ^ 2 + 752 ^2 + 476 ^ 2)^(0.5)). Now, let's look at the next calculations. 

So, now, we're ready to figure out what the moment for the particular problem is. 
Let's first complete the calculation. 

The resultant vector is the sum of all the other forces, and with symbolism we might use R again for the resultant, and the resultant moment of O is the sum of any other moments about O. Forces are sliding vectors. Moments and couples aren't just sliding vectors, they are free vectors, as we call them.

Let's say there's some very simple object and there's some kind of moment that is applied to the object, somehow there is some moment being caused on a particular object, and for whatever reason we know where it is, with a big arm and some type of force. This is equivalent to the same moment/direction/magnitude applied anywhere on that object, since moment itself doesn't have a location. The Force may have a location, but the moment itself doesn't have a specific location to it. 

Let's take this new diagram, and first calculate all of the forces of the diagram windmill turned by a motor. 

When we design the mounting force, we need to oppose this force in order to get static equilibrium.

To calculate the moment based on a force, we need to calculate the location based on the reference point. Moments are free vectors, which don't have a particular location, and as a result you can add the 600Nm with ease. 

Remember, the moment can be applied anywhere. The Moment of a force is a measure of its tendency to cause a body to rotate about a specific point or axis.