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Sunday, 1 March 2020

LAMI'S THEOREM IN ENGINEERING MECHANICS

We were discussing the projectile motion - trajectory equation, definition and formulas and we have also seen the concept of Terminal velocity and apparent weight of a man in a lift with the help of our previous posts.  

Now, we will be interested further to understand a very important topic i.e. Lami's theorem in engineering mechanics. 

We will find out here the statement, explanations, importance, applications and limitations of lami's theorem with the help of this post. We will also see here the Conditions to apply the Lami’s theorem. 


Lami's theorem in engineering mechanics 

Let us first see here what is the statement of Lami's theorem in engineering mechanics.
Lami’s theorem states that 

If three coplanar, concurrent and non-collinear forces are in equilibrium, the magnitude of each force will be proportional to the sine of the angle between the other two forces. 


Conditions to apply the Lami’s theorem 

There are following four conditions, as mentioned below, in order to apply the Lami’s theorem.
We must have to remember that all the three forces acting on a body must be in a same plane.  

We must have to remember that these three forces must be concurrent i.e. these three forces must be acting through a same point. 

We must have to remember that these three forces must be non-collinear i.e. these forces must not be acting in a same line of action. 

We must have to remember that these three coplanar, concurrent and non-collinear forces acting on a body must be in equilibrium i.e. there must not be any acceleration in the body upon the application of these three forces. We can also say that there must be zero net force over the body when these three forces will be applied. 

If these three forces are not following the above mentioned conditions, Lami’s theorem will not be applicable. 


Explanation of Lami’s theorem 

Let us assume that there is an object or a body of any given shape, as mentioned below, are exerted by three forces A, B and C. Angles made by these three forces with each other are displayed here in following figure as α, β and γ. 



According to Lami’s theorem statement, we will have following equation as mentioned here. 

A α Sin α, A = K Sin α
B α Sin β, A = K Sin β
C α Sin γ, A = K Sin γ 

Or we can also redefine it as 

K = A/ Sin α = B/ Sin β = C/ Sin γ 

Therefore, according to Lami’s theorem statement, we will have following equation which could be used in order to secure the magnitude of the unknown force in a case where object will be exerted by three coplanar, concurrent and non-collinear forces and object or body is in equilibrium.  


Limitations of Lami’s theorem 

There are few limitations of Lami’s theorem as mentioned here. 

There should be only three forces acting on the object. If there are more than three forces or less than three forces, we may not apply the Lami’s theorem. 

Lami’s theorem is applicable to only coplanar, concurrent and non-collinear forces. 

Therefore, we have seen here the basics of Lami’s theorem, proof of Lami's theorem and its limitations. 

Further we will find out another concept in engineering mechanics with the help of our next post.   

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Reference:  

Engineering Mechanics, By Prof K. Ramesh  
Image courtesy: Google    

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