Lami's Theorem

This theorem is applicable to three coplanar concurrent forces acting on a body.

"It states that If three coplanar concurrent forces are acting on a body and the body is in equilibrium then each force is proportional to the sine of angle between the other two forces."

It is named after scientist Bernard Lamy.

He was a french mathematician who gave an equation of relation between three coplanar concurrent forces acting on a body.

Note:- This theorem in only applicable to a system of three coplanar concurrent forces. 

Now consider the figure shown below

Three forces namely F1,F2,F3 are acting on a body whose line of action pass through same point and the forces exist in the same plane.

Now say α is the angle between the forces F2 and F3, β is the angle between F1 and F3 and γ is the angle between F1 and F2.

Since the body is in equilibrium so a free body diagram of the body is shown below.

According to Lami's Theorem

`frac{F1}{sinα}`=`frac{F2}{sinbeta}`=`frac{F3}{Singamma}`

For Example consider this question:-

Q. Three forces 60N, 30N and P are acting on a body. If angle between the forces 60N and 30N is 120 and angle between 30N and 20N is 180, find the angle between the other two forces and also find the magnitude of P

Sol:-

Given F1=60N

F2=30N

α = 120,  β = 180 and 𝛄 = 360 - (α + β) = 360-(120+180) = 100

Now, According to lami's Theorem

`frac{F1}{sinα}`=`frac{F2}{sinbeta}`=`frac{F3}{Singamma}`

or ,

`frac{60}{sin120}`=`frac{30}{sin180}`=`frac{P}{Sin100}`

or,

`frac{60}{sin120}`= `frac{P}{Sin100}`

Hence,  P = 68.23 N Ans

Lami's Theorem can be used in solving forces in various engineering and real life problems problems:

For Example

1. Crane lifting an object
2. Swings hanging through a rigid support
3. Caty used to through stones.

and many more.