State and Prove Lami’s Theorem

Lami’s Theorem:

It states that “If three coplanar forces acting at a point be in equilibrium, then each force is proportional to the sine of the angle between the other two”. Mathematically,

P/sinα = Q/sinβ = R/sinγ

where, P, Q and R are three forces and α, β, and γ are the angles as shown in the below figure.

Prove Lami’s Theorem:

Consider three coplanar forces P, Q and R acting at a point O. Let the opposite angles to three forces be α, β and γ as shown in below figure:

Now, let us complete the parallelogram OACB with OA and OB as adjacent sides as shown in the figure. We know that the resultant of two forces P and Q will be given by the diagonal OC both in magnitude and direction of the parallelogram OACB.

Since these forces are in equilibrium, therefore the results of the forces P and Q must be in line with OD and equal to R, but in opposite direction.