3.3: Couples

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A couple is a set of equal and opposite forces that exerts a net moment on an object but no net force. Because the couple exerts a net moment without exerting a net force, couples are also sometimes called pure moments.

A lugwrench being used to tighten/remove a lug nut on an automobile wheel. Vectors are depicted for the rotational forces exerted in opposite directions on the two sides of the wrench handle. — Figure \(\PageIndex{1}\): The two equal and opposite forces exerted on this lug wrench are a couple. They exert a moment on the lug nut on this wheel without exerting any net force on the wheel. Adapted from image by Steffen Heinz Caronna CC-BY-SA 3.0.

The moment exerted by a couple also differs from the moment exerted by a single force in that it is independent of the location you are taking the moment about. In the example below we have a couple acting on a beam. Each force has a magnitude \(F\) and the distance between the two forces is \(d\).

A horizontal beam has forces of equal magnitude but opposite direction (up vs down) exerted upon it at different points along its length. There is a distance x from one end of the beam to the point of application of the closer force, and a distance d between the points of application of the two forces. — Figure \(\PageIndex{2}\): The moment exerted by this couple is independent of the of the distance \(x\).

Now we have some point A, which is distance \(x\) from the first of the two forces. If we take the moment of each force about point A, and then add these moments together for the net moment about A we are left with the following formula.

\[ M \, = \, -(F*x) + (F*(x+d)) \] If we rearrange and simplify the formula above, we can see that the variable \(x\) actually disappears from the equation, leaving the net moment equal to the magnitude of the forces (\(F\)) times the distance between the two forces (\(d\)).

\[ M \, = \, -(F*x) + (F*x) + (F*d) \]

\[ M \, = \, (F*d) \]

This means that no matter what value of \(x\) we have, the magnitude of the moment exerted by the couple will be the same. The magnitude of the moment due to the couple is independent of the location we are taking the moment about. This will work in two or three dimensions as well. The magnitude of the moment due to a couple will always be equal to the magnitude of the forces times the perpendicular distance between the two forces.

Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/2U4APUz__Gk.

Example \(\PageIndex{1}\)

What is the moment that the couple below exerts about point A?

A horizontal rod has a point near its center designated A. An upward force of 60 lbs is applied to the rod 3 feet to the right of A, and a downard force of 60 lbs is applied to the rod 3 feet to the left of A. — Figure \(\PageIndex{3}\): problem diagram for Example \(\PageIndex{1}\). A couple is applied to a rod, with each force being equidistant from point A on the rod.

Solution: Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/rfD-b6V5qNY.

Example \(\PageIndex{2}\)

What is the moment that the couple below exerts about point A?

A horizontal rod has a central point designated as point A. A force of magnitude 3 kiloNewtons, pointing down and to the right, is applied to the rod at point A; another force of the same magnitude but in the opposite direction is applied to the rod 1.5 meters to the right of A. This force makes a 60-degree angle with the vertical. — Figure \(\PageIndex{4}\): problem diagram for Example \(\PageIndex{2}\). A couple is applied to a rod, with one force applied directly at point A and the other being applied 1.5 meters to the right of A.

Solution: Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/s3-RJwKkH3Y.