Lami's theorem

Lami's theorem In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. D'après le théorème, {style d'affichage {frac {UN}{sin alpha }}={frac {B}{sin beta }}={frac {C}{sin gamma }}} où un, B and C are the magnitudes of the three coplanar, concurrent and non-collinear vectors, {style d'affichage V_{UN},V_{B},V_{C}} , which keep the object in static equilibrium, and α, β and γ are the angles directly opposite to the vectors.[1] Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.[2] Contenu 1 Preuve 2 Voir également 3 Références 4 Further reading Proof As the vectors must balance {style d'affichage V_{UN}+V_{B}+V_{C}=0} , hence by making all the vectors touch its tip and tail we can get a triangle with sides A,B,C and angles {displaystyle 180^{o}-alpha ,180^{o}-bêta ,180^{o}-gamma } . By the law of sines then[1] {style d'affichage {frac {UN}{péché(180^{o}-alpha )}}={frac {B}{péché(180^{o}-bêta )}}={frac {C}{péché(180^{o}-gamma )}}.} Then by applying that for any angle {thêta de style d'affichage } , {péché de style d'affichage(180^{o}-thêta )=sin theta } on obtient {style d'affichage {frac {UN}{sin alpha }}={frac {B}{sin beta }}={frac {C}{sin gamma }}.} See also Mechanical equilibrium Parallelogram of force Tutte embedding References ^ Jump up to: a b Dubey, N. H. (2013). Engineering Mechanics: Statics and Dynamics. Tata McGraw-Hill Education. ISBN 9780071072595. ^ "Lami's Theorem - Oxford Reference". Récupéré 2018-10-03. Further reading R.K. Bansal (2005). "A Textbook of Engineering Mechanics". Laxmi Publications. p. 4. ISBN 978-81-7008-305-4. I.S. Gujral (2008). "Engineering Mechanics". Firewall Media. p. 10. ISBN 978-81-318-0295-3 Catégories: StaticsPhysics theorems