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,

$\frac{v_A}{\sin \alpha }=\frac{v_B}{\sin \beta }=\frac{v_C}{\sin \gamma }$

where $v_A,v_B,v_C$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors, ${\overrightarrow{v}}_A,{\overrightarrow{v}}_B,{\overrightarrow{v}}_C$, which keep the object in static equilibrium, and $\alpha ,\beta ,\gamma$ are the angles directly opposite to the vectors, thus satisfying $\alpha +\beta +\gamma =360^o$.

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 ${\overrightarrow{v}}_A+{\overrightarrow{v}}_B+{\overrightarrow{v}}_C=\overrightarrow{0}$, hence by making all the vectors touch its tip and tail the result is a triangle with sides $v_A,v_B,v_C$ and angles $180^o-\alpha ,180^o-\beta ,180^o-\gamma$ ($\alpha ,\beta ,\gamma$ are the exterior angles).

By the law of sines then

$\frac{v_A}{\sin (180^o-\alpha )}=\frac{v_B}{\sin (180^o-\beta )}=\frac{v_C}{\sin (180^o-\gamma )}.$

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

$\frac{v_A}{\sin \alpha }=\frac{v_B}{\sin \beta }=\frac{v_C}{\sin \gamma }.$

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

Further reading

• R.K. Bansal (2005). "A Textbook of Engineering Mechanics". Laxmi Publications. p. 4. Template:ISBN.
• I.S. Gujral (2008). "Engineering Mechanics". Firewall Media. p. 10. Template:ISBN