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. According to the theorem, {displaystyle {frac {A}{sin alpha }}={frac {B}{sin beta }}={frac {C}{sin gamma }}} where A, B and C are the magnitudes of the three coplanar, concurrent and non-collinear vectors, {displaystyle V_{A},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] Contents 1 Proof 2 See also 3 References 4 Further reading Proof As the vectors must balance {displaystyle V_{A}+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}-beta ,180^{o}-gamma } . By the law of sines then[1] {displaystyle {frac {A}{sin(180^{o}-alpha )}}={frac {B}{sin(180^{o}-beta )}}={frac {C}{sin(180^{o}-gamma )}}.} Then by applying that for any angle {displaystyle theta } , {displaystyle sin(180^{o}-theta )=sin theta } we obtain {displaystyle {frac {A}{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". Retrieved 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 Categories: StaticsPhysics theorems