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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. | |||
:<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 18:30, 28 January 2014
In statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and 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.
Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after 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 triangle law, we can re-construct the diagram as follow:
By the law of sines,
See also
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