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,

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

where A, B and C are the magnitudes of the three coplanar, concurrent and non-collinear vectors, ${\displaystyle V_{A},V_{B},V_{C}}$, which keep the object in static equilibrium, and α, β and γ are the angles directly opposite to the vectors.[1]

Illustration of Lami's theorem

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 ${\displaystyle V_{A}+V_{B}+V_{C}=0}$, hence by making all the vectors touch its tip and tail we can get a triangle with sides A,B,C and angles ${\displaystyle 180^{o}-\alpha ,180^{o}-\beta ,180^{o}-\gamma }$. By sine rule.[1]

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

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

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

References

1. Dubey, N. H. (2013). Engineering Mechanics: Statics and Dynamics. Tata McGraw-Hill Education. ISBN 9780071072595.
2. "Lami's Theorem - Oxford Reference". Retrieved 2018-10-03.

Further reading

• R.K. Bansal (2005). "A Textbook of Engineering Mechanics". Laxmi Publications. p. 4. ISBN 978-81-7008-305-4.
• I.S. Gujral (2008). "Engineering Mechanics". Firewall Media. p. 10. ISBN 978-81-318-0295-3