Couples
A couple is a pair of equal and opposite forces that twist without pushing. Couples are how steering wheels, screwdrivers, and taps work.
Definition
A couple consists of two parallel forces with equal magnitudes, opposite directions, and separate lines of action. Their resultant force is zero, so they cannot move a body's centre, but they produce a pure turning effect, a couple moment.
Couple moment
The moment of a couple is \(M = Fd\), where d is the perpendicular distance between the two forces' lines of action. Turning a steering wheel 0.4 m across with 25 N from each hand produces a couple of 25 x 0.4 = 10 N·m.
Same about every point
Unlike a single force, a couple has the same moment about every point in the plane. Taking moments about any point, the distances to the two forces always differ by d, so the result is always Fd. A couple moment is therefore a free vector: it can be moved anywhere on the body without changing its effect.
Equivalent couples
Two couples are equivalent if they have the same moment. A couple of 50 N with arm 0.2 m (10 N·m) is equivalent to 100 N with arm 0.1 m. A wider grip lets you produce the same couple with less force, which is why large taps and valve wheels are easier to turn.
Adding couples
Couple moments add like vectors, or as signed numbers in 2D. A shaft receiving 30 N·m anticlockwise from a motor and 12 N·m clockwise from friction experiences a net couple of 18 N·m anticlockwise.
Couples in 3D
In three dimensions, a couple moment is a vector perpendicular to the plane of the two forces, with \(\mathbf{M} = \mathbf{r} \times \mathbf{F}\), where r goes from any point on one force's line to any point on the other's. Couple vectors in different directions combine by vector addition.
Couples in engineering
Engines deliver torque to driveshafts as couples. Gearboxes change the couple's magnitude while the transmitted power stays roughly constant. Screwdrivers with fatter handles let you apply larger couples with the same grip force, and wrenches with two-handed T-bars apply nearly pure couples to wheel nuts.
Worked example
A plumber turns a T-handle valve key whose handle is 0.3 m wide, pushing 60 N with each hand in opposite directions. The couple is 60 x 0.3 = 18 N·m. The valve's resistance is 25 N·m, so it will not turn; to reach 25 N·m with the same 60 N grip, the handle would need to be about 0.42 m wide, or each hand would need about 83 N.
Key takeaways
A couple is two equal, opposite, parallel forces with separate lines of action; its resultant force is zero. Its moment is M = Fd, d being the perpendicular distance between the forces, and it is the same about every point, so a couple moment is a free vector. Couples with equal moments are equivalent, so wider grips need less force. Couple moments add as vectors. In 3D, M = r x F with r between the two lines. Engines, steering wheels, and valve keys all apply couples.
Flashcard
Equal, opposite, parallel forces apart
Check your understanding
Hands apply 25 N each, in opposite directions, on a steering wheel 0.4 m across. What is the couple?