伐里农定理

证明

考虑 N 个力矢 f ₁,f ₂,...,f _(N)，它们共同作用于一点 O，则结果为： F =∑ ᵢ₌₁^(N)f ᵢ.

每个分力相对于其他点 O ₁ 的力矩为 （公式略）.

将力矩相加并去掉公因子 （公式略），可见结果可完全用 F 表示，实际上就是 F 相对于 O ₁ 点的力矩： （公式略）.

证明了定理，即关于 O ₁ 的合力矩与各分力的分力矩之和相同。

Varignon's theorem is a theorem of French mathematician Pierre Varignon (1654–1722), published in 1687 in his book Projet d'une nouvelle mécanique. The theorem states that the torque of a resultant of two concurrent forces about any point is equal to the algebraic sum of the torques of its components about the same point.

In other words, "If many concurrent forces are acting on a body, then the algebraic sum of torques of all the forces about a point in the plane of the forces is equal to the torque of their resultant about the same point."

Proof

Consider a set of N force vectors f ₁,f ₂,...,f _(N) that concur at a point O in space. Their resultant is: F =∑ ᵢ₌₁^(N)f ᵢ.

The torque of each vector with respect to some other point O ₁ （公式略）.

Adding up the torques and pulling out the common factor （公式略）, one sees that the result may be expressed solely in terms of F, and is in fact the torque of F with respect to the point O ₁: （公式略）.

Proving the theorem, i.e. that the sum of torques about O ₁ is the same as the torque of the sum of the forces about the same point.