VARIGNON’S THEOREM (OR PRINCIPAL OF MOMENTS)Varignon’s Theorem states that the moment of a force about any point is equal to the algebraic sum of the moments of its components about that point. Principal of moments states that the moment of the resultant of a number of forces about any point is equal to the algebraic sum of the moments of all the forces of the system about the same point. Proof of Varignon’s Theorem. Fig. 2. 3. 4 (a)Fig. Fig. 2. 3. 4 (a) shows two forces Fj and F2 acting at point O. These forces are represented in magnitude and direction by OA and OB. Their resultant R is represented in magnitude and direction by OC which is the diagonal of parallelogram OACB. THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE. Theorem of projections ««««««««««« 18 2.7. Sample problems «««««««« «««««««««« 19 2.8. Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram. The proof applies even to skew quadrilaterals in spaces of any dimension. Moment of a force about any point is equal to the sum of the moments of the components of that force about the same point. To prove this theorem, consider the force R acting in the plane of the body shown. The Principle of Moments, also known as Varignon's Theorem, states that the moment of any force is equal to the algebraic sum of the moments of the components of that force. Parallelogram is formed. Varignon’s proof was pub-lished in 1731 in Elemens de Mathematique, a post - humously published volume of mathematical con-. Pierre Varignon and the Parallelogram Theorem S This article is the. Let O’ is the point in the plane about which moments of F1, F2and Rare to be determined. From point O’, draw perpendicularson OA, OC and OB. Let r. 1= Perpendicular distance between F1 and O’.
Join OO’ and produce it to D. From points C, A and B draw perpendiculars on OD meeting at D, E and F respectively. From A and B also draw perpendiculars on CD meeting the line CD at G and H respectively. Let . 2. 3. 4 (b), OA = BC and also OA parallel to BC, hence the projection of OA and BC on the same vertical line CD will be equal i. GD = CH as GD is the projection of OA on CD and CH is the projection of BC on CD. P1 sin . Hence projections of OB and AC on the same horizontal line OD will be equal i. OF = ED) R sin . Hence moment of R about any point in the algebraic sum of moments of its components i. F1and F2)about the same point. Hence Varignon’s principle is proved. The principle of moments (or Varignon’s principle) is not restricted to only two concurrent forces but is also applicable to any coplanar force system, i. Problem 2. 1. 7 A force of 1. N is acting at a point A as shown in Fig. Determine the moments of this force about O. Sol. Given: Force at A = 1. NDraw a perpendicular from O on the line of action of force 1. N. Hence OB is the perpendicular on the line of action of 1. N as shown in Fig. Method Triangle OBC is a right- angled triangle. And angle. OCB= 6. Sin. 60o = 0. B/OCOB = OC sin 6. Ans. 2nd Method The moment of force 1. N about O, can also be determined by using Varignon’s principle. The force 1. 00 N is replaced by its two rectangular components at any convenient point. Here the convenient point is chosen as C. The horizontal and vertical components of force 1. N acting at C are shown in Fig. The horizontal component= 1. Moment of this force about O=8.
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