Physics: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,
- [math]\displaystyle{ \frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma} }[/math]
where [math]\displaystyle{ v_A, v_B, v_C }[/math] are the magnitudes of the three coplanar, concurrent and non-collinear vectors, [math]\displaystyle{ \vec{v}_A, \vec{v}_B, \vec{v}_C }[/math], which keep the object in static equilibrium, and [math]\displaystyle{ \alpha,\beta,\gamma }[/math] are the angles directly opposite to the vectors,[1] thus satisfying [math]\displaystyle{ \alpha+\beta+\gamma=360^o }[/math].
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 [math]\displaystyle{ \vec{v}_A+\vec{v}_B+\vec{v}_C=\vec{0} }[/math], hence by making all the vectors touch its tip and tail the result is a triangle with sides [math]\displaystyle{ v_A,v_B,v_C }[/math] and angles [math]\displaystyle{ 180^o -\alpha, 180^o -\beta, 180^o -\gamma }[/math] ([math]\displaystyle{ \alpha,\beta,\gamma }[/math] are the exterior angles).
By the law of sines then[1]
[math]\displaystyle{ \frac{v_A}{\sin (180^o -\alpha)}=\frac{v_B}{\sin (180^o-\beta)}=\frac{v_C}{\sin (180^o-\gamma)}. }[/math]
Then by applying that for any angle [math]\displaystyle{ \theta }[/math], [math]\displaystyle{ \sin (180^o - \theta) = \sin \theta }[/math] (suplementary angles have the same sine), and the result is
[math]\displaystyle{ \frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}. }[/math]
See also
References
- ↑ 1.0 1.1 Dubey, N. H. (2013) (in en). Engineering Mechanics: Statics and Dynamics. Tata McGraw-Hill Education. ISBN 9780071072595. https://books.google.com/books?id=8Yf0AQAAQBAJ&q=lamis+theorem#q=lamis%20theorem.
- ↑ "Lami's Theorem - Oxford Reference". http://www.oxfordreference.com/view/10.1093/oi/authority.20110803100049237.
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
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