In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear force vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,
v A sin α = v B sin β = v C sin γ {\displaystyle {\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}}
where v A , v B , v C {\displaystyle v_{A},v_{B},v_{C}} are the magnitudes of the three coplanar, concurrent and non-collinear vectors, v → A , v → B , v → C {\displaystyle {\vec {v}}_{A},{\vec {v}}_{B},{\vec {v}}_{C}} , which keep the object in static equilibrium, and α , β , γ {\displaystyle \alpha ,\beta ,\gamma } are the angles directly opposite to the vectors, thus satisfying α + β + γ = 360 o {\displaystyle \alpha +\beta +\gamma =360^{o}} . Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.
Proof As the vectors must balance v → A + v → B + v → C = 0 → {\displaystyle {\vec {v}}_{A}+{\vec {v}}_{B}+{\vec {v}}_{C}={\vec {0}}} , hence by making all the vectors touch its tip and tail the result is a triangle with sides v A , v B , v C {\displaystyle v_{A},v_{B},v_{C}} and angles 180 o − α , 180 o − β , 180 o − γ {\displaystyle 180^{o}-\alpha ,180^{o}-\beta ,180^{o}-\gamma } ( α , β , γ {\displaystyle \alpha ,\beta ,\gamma } are the exterior angles). By the law of sines then
v A sin ( 180 o − α ) = v B sin ( 180 o − β ) = v C sin ( 180 o − γ ) . {\displaystyle {\frac {v_{A}}{\sin(180^{o}-\alpha )}}={\frac {v_{B}}{\sin(180^{o}-\beta )}}={\frac {v_{C}}{\sin(180^{o}-\gamma )}}.}
Then by applying that for any angle θ {\displaystyle \theta } , sin ( 180 o − θ ) = sin θ {\displaystyle \sin(180^{o}-\theta )=\sin \theta } (supplementary angles have the same sine), and the result is
v A sin α = v B sin β = v C sin γ . {\displaystyle {\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}.}
See also Mechanical equilibrium Parallelogram of force Tutte embedding
References
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