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Varignon's theorem is a theorem of French mathematician
Pierre Varignon Pierre Varignon (; 1654 – 23 December 1722) was a French mathematician. He was educated at the Society of Jesus, Jesuit College and the University of Caen, where he received his Magister Artium, M.A. in 1682. He took Holy Orders the following ...
(1654–1722), published in 1687 in his book ''Projet d'une nouvelle mécanique''. The theorem states that the
torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
of a
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
of two concurrent forces about any point is equal to the
algebraic sum In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynom ...
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 vector In physics, a force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the magnitude and direction ...
s \mathbf_1, \mathbf_2, ..., \mathbf_N that concur at a point \mathbf in space. Their resultant is: :\mathbf=\sum_^N \mathbf_i . The torque of each vector with respect to some other point \mathbf_1 is : \mathbf_^ = (\mathbf-\mathbf_1) \times \mathbf_i . Adding up the torques and pulling out the common factor (\mathbf-\mathbf), one sees that the result may be expressed solely in terms of \mathbf , and is in fact the torque of \mathbf with respect to the point \mathbf_1: : \sum_^N \mathbf_^ = (\mathbf-\mathbf_1) \times \left ( \sum_^N \mathbf_ \right ) = (\mathbf-\mathbf_1) \times \mathbf = \mathbf_^ . Proving the theorem, i.e. that the sum of torques about \mathbf_1 is the same as the torque of the sum of the forces about the same point.


References


External links


Varignon's Theorem
at
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Eponymous theorems of physics Mechanics Moment (physics) {{classicalmechanics-stub