4.2: Resolution of a Force into a Force and a Couple

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As discussed on the sections on moments, a force can have a tendency to cause both a linear and angular acceleration. For example, below is diagram of a force acting on an extended body. If we were to think about everything relative to the center of mass of this body, there would be some force acting on the center of mass that would cause the same linear acceleration and some pure moment (couple) that would cause the same angular acceleration. This would be a force and couple that is statically equivalent to the original force, though the point of application of the force changes.

A rectangular body experiences two forces: Force A is applied at point A, the rectangle's lower right corner, and points upwards. Force B is applied at point B, the center of the rectangle, and possesses the same magnitude and direction as Force A. A counterclockwise moment vector, Moment B, is drawn around point B. — Figure \(\PageIndex{1}\): The original force acting at A would cause both linear and angular acceleration. The force at B would cause the same linear acceleration and the moment at B would cause the same angular acceleration. The force and the moment at B are statically equivalent to the original force at A.

The process of transforming one force applied at one point, into a force and a couple at some other point is known as resolving a force into a force and a couple. There are a few reasons that we may want to do this, but one primary reason is to find the equivalent force couple system for a complex set of forces and moments. The equivalent force couple system is used to simplify more complex analysis, and consists of a single force and a single pure moment (couple) that are statically equivalent to some more complex combination of forces and moments. An important first step in finding the equivalent force couple system is to resolve all the forces so that everything is acting at the same point.

In order to visualize the process of resolving a force into a force and a couple, you can use the process outlined in the diagram below. Imagine we have a body with a force acting at some point A. We want to resolve the force into a force and a couple about some other point B. To do this we will first add two forces to the diagram at point B. One will have the same magnitude and direction as the original force and the other will be equal and opposite to the original force. Because these two forces are equal, opposite, and collinear, this will not change the situation (it's the equivalent to adding zero to an equation). Now, with these three forces acting on the diagram, we can break it down into two sets. The first is a force acting at point B with the same magnitude and direction as the original force. The other two forces act as a couple, exerting a pure moment about point B. Finally, we can redraw the system as a force acting at point B and the pure moment acting about point B.

A rectangle has a force A applied at point A, its lower right corner; another point B is marked at the rectangle center. First, two equal and opposite forces are applied at B, with one of these forces being the same in magnitude and direction as force A. Force A and its opposite force at B are the couple component, exerting a pure moment about point B; the force at B with the same direction as force A is the force component. Finally, the two couple component vectors are removed from the diagram and replaced with a moment vector about point B. — Figure \(\PageIndex{2}\): The process of resolving a force about some point A into a force and a couple about some point B.

As a shortcut to the process described above, we can also see that the force in the equivalent force couple system will always have a magnitude and direction equal to the original force and the couple will be equal to the moment exerted by the original force about the new point of application.

Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/sLixBOWBCuY.

Example \(\PageIndex{1}\)

Resolve the force shown below to a force and a couple acting at point A.

A horizontal rod with its left end marked as point A. An upwards force of magnitude 4 lbs is applied to the rod at a point 6 feet to the right of A. — Figure \(\PageIndex{3}\): problem diagram for Example \(\PageIndex{1}\); a rod experiences a single, upwards force applied 6 feet from the left end of the rod (point A).

Solution: Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/Wfl3S_5FD0Q.

Example \(\PageIndex{2}\)

Resolve the force shown below to a force and a couple acting at point A.

A horizontal bar with the left end marked as point A. A force with magnitude 55 kiloNewtons, pointing down and to the right to make a 55-degree angle with the vertical, is applied to the rod at a point 0.6 meters to the right of point A. — Figure \(\PageIndex{4}\): problem diagram for Example \(\PageIndex{2}\); a rod experiences a single force, pointing downwards and to the right, applied 0.6 meters from the left end of the rod (point A).

Solution: Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/1hRrKhWzf98.