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,

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

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

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.

Proof

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

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

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

See also

• Mechanical equilibrium
• Parallelogram of force
• Tutte embedding

Further reading

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