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In [[statics]], '''Lami's theorem''' is an equation relating the magnitudes of three [[coplanar]], [[Concurrent lines|concurrent]] and [[Coplanarity|non-collinear]] forces, which keeps an object in [[static equilibrium]], with the angles directly opposite to the corresponding forces. A,B,C


:where ''A'', ''B'' and ''C'' are the magnitudes of three coplanar, concurrent and non-collinear forces, which keep the object in static equilibrium, and


:''α'', ''β'' and ''γ'' are the angles directly opposite to the forces ''A'', ''B'' and ''C'' respectively.
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:<math>\Rightarrow \frac{A}{\sin \alpha}=\frac{B}{\sin \beta}=\frac{C}{\sin \gamma}</math>
 
:[[File:Lami.png|alt=Lami's Theorem]]
 
Lami's theorem is applied in static analysis of mechanical and structural systems.  The theorem is named after [[Bernard Lamy (mathematician)|Bernard Lamy]].
 
==Proof of Lami's Theorem==
Suppose there are three coplanar, concurrent and non-collinear forces, which keeps the object in static equilibrium. By the [[Euclidean vector|triangle law]], we can re-construct the diagram as follow:
 
:[[File:LamiProof.png]]
 
By the [[law of sines]],
:<math>\frac{A}{\sin (\pi - \alpha)}=\frac{B}{\sin (\pi - \beta)}=\frac{C}{\sin (\pi - \gamma)}</math>
 
:<math>\Rightarrow \frac{A}{\sin \alpha}=\frac{B}{\sin \beta}=\frac{C}{\sin \gamma}</math>
 
==See also==
* [[Law of sines]]
* [[Mechanical equilibrium]]
* [[Bernard Lamy (mathematician)]]
 
==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
 
[[Category:Statics]]
[[Category:Physics theorems]]

Revision as of 01:27, 27 February 2014


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