Couple of forces

Japanese: 偶力 - ぐうりょく

When two forces of equal magnitude, parallel, and opposite directions act on different points of application of an object, these two forces are treated as a pair and are called a couple. This pair of forces does not change the motion of the object's center of gravity, but only causes the object to rotate. In general, the motion of a rigid body (a hard object with an expansive surface) can be divided into the motion of the center of gravity and the rotational motion around the center of gravity. The former is governed by the sum of the vectors of the forces acting on each point of the rigid body, while the latter is governed by the rotational ability of each force, i.e., the sum of the rotational moments. In the case of a couple, the vector sum of the two forces f ₁ and f ₂ , f ₁ + f ₂ , is zero because f ₂ = - f ₁ , and the motion of the center of gravity remains unchanged. Therefore, when a couple acts on a stationary rigid body, the object as a whole does not move. However _, the sum of _the rotational moments is calculated by vector multiplication of two position vectors r1 _and r2 as r1 _× _f1 + r2 × f2 ₌ ₍ r1 - r2 ) _× f1 _.
This will not be zero unless the relative position vector r ₁ - r ₂ is parallel to f ₁ ( see figure ). In the parallel case, the lines of action of the two forces coincide.

In general, the magnitude of the rotational moment of a couple is given by their arithmetic product fl, where l is the distance between the parallel lines of action of the two forces, _{and |f1} | = | f2 | = f _, according to the definition of the vector product, and does not depend on the point of application of each force. l is called the arm length of the couple, and corresponds to the arm length of the lever principle.

[Yasuhisa Abe]

Couple (Diagram)
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Couple (Diagram)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

大きさが等しく、互いに平行で、反対向きの二つの力が一つの物体の異なる作用点に働くとき、この二つの力を1対として取り扱い、偶力という。この1対の力は物体の重心の運動を変化させず、物体の回転のみを引き起こす。一般に剛体（広がりのある固い物体）の運動は、重心の運動と重心の周りの回転運動に分けられる。前者は剛体の各点に働く力のベクトルの和によって支配され、後者はそれぞれの力の回転させる能力、すなわち回転のモーメントの和によって支配される。偶力の場合、二つの力f₁とf₂のベクトルの和f₁＋f₂は、f₂＝－f₁であるからゼロとなり、重心の運動は不変である。したがって、静止している剛体に偶力が作用した場合には、全体として物体が移動することはない。しかし、回転モーメントの和は、二つの位置ベクトルをr₁、r₂としたとき、ベクトル積を用いて
　r₁×f₁＋r₂×f₂＝(r₁－r₂)×f₁
となり、相対位置ベクトルr₁－r₂がf₁と平行でない限りゼロにならない（図）。平行の場合は二つの力の作用線が一致する。