Lami's theorem explained

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,

(v _{A})/(= \sin \alpha) (v _{B})/(= \sin \beta) (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,^[1] 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.^[2]

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^[1]

(v _{A})/(= \sin ( 180 ^{o} - \alpha)) (v _{B})/(= \sin ( 180 ^{o-\beta)}) (v _{C})/(\sin ( 180 ^{o-\gamma)}) .

Then by applying that for any angle

\theta , \sin ( 180 ^{o} - \theta) = \sin \theta (suplementary angles have the same sine), and the result is(v _{A})/(= \sin \alpha) (v _{B})/(= \sin \beta) (v _{C})/(\sin \gamma) .

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

References

1. Book: Dubey, N. H.. Engineering Mechanics: Statics and Dynamics. 2013. Tata McGraw-Hill Education. 9780071072595. en.
2. Web site: Lami's Theorem - Oxford Reference. 2018-10-03.

Further reading

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