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,

${\displaystyle {\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}}$

where ${\displaystyle v_{A},v_{B},v_{C}}$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors, ${\displaystyle {\vec {v}}_{A},{\vec {v}}_{B},{\vec {v}}_{C}}$, which keep the object in static equilibrium, and ${\displaystyle \alpha ,\beta ,\gamma }$ are the angles directly opposite to the vectors,^[1] thus satisfying ${\displaystyle \alpha +\beta +\gamma =360^{o}}$.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.^[2]

Proof

As the vectors must balance ${\displaystyle {\vec {v}}_{A}+{\vec {v}}_{B}+{\vec {v}}_{C}={\vec {0}}}$, hence by making all the vectors touch its tip and tail the result is a triangle with sides ${\displaystyle v_{A},v_{B},v_{C}}$ and angles ${\displaystyle 180^{o}-\alpha ,180^{o}-\beta ,180^{o}-\gamma }$ (${\displaystyle \alpha ,\beta ,\gamma }$ are the exterior angles).

By the law of sines then^[1]

${\displaystyle {\frac {v_{A}}{\sin(180^{o}-\alpha )}}={\frac {v_{B}}{\sin(180^{o}-\beta )}}={\frac {v_{C}}{\sin(180^{o}-\gamma )}}.}$

Then by applying that for any angle ${\displaystyle \theta }$, ${\displaystyle \sin(180^{o}-\theta )=\sin \theta }$ (supplementary angles have the same sine), and the result is

${\displaystyle {\frac {v_{A}}{\sin \alpha }}={\frac {v_{B}}{\sin \beta }}={\frac {v_{C}}{\sin \gamma }}.}$

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding