Moment About a Point (Scalar Calculations)
Given any point on an extended body, if there is a force acting on that body that does not travel through that point, then that force will cause a moment about that point. As discussed on the moments page, a moment is a force's tendency to cause rotation.
The Scalar Method in 2 Dimensions
In discussing how to calculate the moment of a force about a point via scalar quantities, we will begin with the example of a force on a simple lever as shown below. In this simple lever there is a force on the end of the lever, distance d away from the center of rotation for the lever (point A) where the force has a magnitude F.
When using scalar quantities, the magnitude of the moment will be equal to the perpendicular distance between the line of action of the force and the point we are taking the moment about.
To determine the sign of the moment, we determine what type of rotation the force would cause. In this case we can see that the force would cause the lever to rotate counterclockwise about point A. Counterclockwise rotations are caused by positive moments while clockwise rotations are caused by negative moments.
Another important factor to remember is that the value d is the perpendicular distance from the force to the point we are taking the moment about. We could measure the distance from point A to the head of the force vector, or the tail of the force vector, or really any point along the line of action of force F. The distance we need to use for the scalar moment calculation however is the shortest distance between the point and the line of action of the force. This will always be a line perpendicular to the line of action of the force, going to the point we are taking the moment about.
The Scalar Method in 3 Dimensions
For three dimensional scalar calculations, we will still find the magnitude of the moment in the same way, multiplying the magnitude of the force by the perpendicular distance between the point and the line of action of the force. This perpendicular distance again is the minimum distance between the point and the line of action of the force. In some cases, finding this distance may be difficult.
Another difficult factor in three dimensional scalar problems is finding the axis of rotation, as this is now more complex that just 'clockwise or counterclockwise'. The axis of rotation will be a line traveling though the point we are taking the moment about that is perpendicular to both the force force vector and the perpendicular distance vector. Finding this direction may be quite difficult for more complex problems where the force and/or distance vectors don't line up with a single coordinate direction.
To further find the direction of the moment vector (which way along the established line) we will use the right hand rule. Wrap the fingers of your right hand around the established line with your fingertips curling in the direction the body would rotate. If you do this, your thumb should point out along the line in the direction of the moment vector.