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.[1]

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 $V_{A}+V_{B}+V_{C}=0$ , hence by making all the vectors touch its tip and tail the result is a triangle with sides A, B, C and angles

$180^{o}-\alpha ,180^{o}-\beta ,180^{o}-\gamma .$

By the law of sines then[1]

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

Then by applying that for any angle $\theta$ , $\sin(180^{o}-\theta )=\sin \theta$ , and the result is

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

References

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

Lami's theorem

Proof

See also

References

Further reading