In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
Lami's theorem is an equation relating the magnitudes of three
coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...
,
concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
and
non-collinear vectors, which keeps an object in
static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,
:
where ''A'', ''B'' and ''C'' are the magnitudes of the three coplanar, concurrent and non-collinear vectors,
, which keep the object in static equilibrium, and ''α'', ''β'' and ''γ'' are the angles directly opposite to the vectors.
:
Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after
Bernard Lamy
Bernard Lamy (15 June 1640 – 29 January 1715) was a French Oratorian, mathematician and theologian.
Life
Lamy was born in Le Mans, France. After studying there, he went to join the Maison d'Institution in Paris, and to Saumur thereafter. I ...
.
Proof
As the vectors must balance
, hence by making all the vectors touch its tip and tail we can get a triangle with sides A,B,C and angles
. By
the law of sines then
Then by applying that for any angle
,
we obtain
See also
*
Mechanical equilibrium
In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero ...
*
Parallelogram of force
The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object.
When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. Forces, ...
*
Tutte embedding
References
Further reading
* R.K. Bansal (2005). "A Textbook of Engineering Mechanics". Laxmi Publications. p. 4. .
* I.S. Gujral (2008). "Engineering Mechanics". Firewall Media. p. 10. {{ISBN, 978-81-318-0295-3
Statics
Physics theorems