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, $\vec{v}_A, \vec{v}_B, \vec{v}_C$ , which keep the object in static equilibrium, and $\alpha,\beta,\gamma$ are the angles directly opposite to the vectors,{{Cite book|url=https://books.google.com/books?id=8Yf0AQAAQBAJ&q=lamis+theorem|title=Engineering Mechanics: Statics and Dynamics|last=Dubey|first=N. H.|date=2013|publisher=Tata McGraw-Hill Education|isbn=9780071072595|language=en}} 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.{{Cite web|url=http://www.oxfordreference.com/view/10.1093/oi/authority.20110803100049237|title=Lami's Theorem - Oxford Reference|access-date=2018-10-03}}

Proof

As the vectors must balance $\vec{v}_A+\vec{v}_B+\vec{v}_C=\vec{0}$ , hence by making all the vectors touch its tip and tail the result is a triangle with sides $v_A,v_B,v_C$ 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

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

Lami's theorem

Proof

See also

References

Further reading