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.

Moment About A Point
The magnitude of the moment that force F exerts about point A in this lever will be equal to the magnitude of the force times distance d.

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.

Moment Equation

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.

Moment About a Point
Distance d always needs to be the shortest length between the line of action of the force and the point we are taking the moment about. This distance will be perpendicular to the line of action of the force.

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.

Moments in 3 dimensions
For moments in three dimensions, the moment vector will always be perpendicular to both the force vector F and the distance vector d.
The Right Hand Rule
To use the right hand rule, align your right hand as shown so that your thumb lines up with the axis of rotation for the moment and your curled fingers point in the direction of rotation for your moment. If you do this, your thumb will be pointing in the direction of the moment vector.

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.

Worked Problems:

Question 1:

What is the moment that Force A exerts about point A? What is the moment that Force B exerts about Point A?

Problem 1 Diagram

Solution:



Question 2:

What is the moment that this force exerts about point A? What is the moment this force exerts about point B?

Problem 2 Diagram

Solution:



Question 3:

What are the moments that each of the three tension forces exert about point A (the point where the beams come together)?

Problem 3 Diagram

Solution: