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,

<math>\frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}</math>

where <math>v_A, v_B, v_C</math> are the magnitudes of the three coplanar, concurrent and non-collinear vectors, <math>\vec{v}_A, \vec{v}_B, \vec{v}_C</math>, which keep the object in static equilibrium, and <math>\alpha,\beta,\gamma</math> are the angles directly opposite to the vectors,<ref name=":0">Template:Cite book</ref> thus satisfying <math>\alpha+\beta+\gamma=360^o</math>.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Proof

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

By the law of sines then<ref name=":0"/>

<math>\frac{v_A}{\sin (180^o -\alpha)}=\frac{v_B}{\sin (180^o-\beta)}=\frac{v_C}{\sin (180^o-\gamma)}.</math>

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

<math>\frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}.</math>

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

References

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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