State and prove Lami’s Theorem.

Lami's Theorem is a simple and useful theorem in statics, a branch of physics that deals with forces and their effects on bodies in equilibrium.

Statement of Lami's Theorem: If a body is in equilibrium under the action of three forces, then each force is proportional to the sine of the angle between the other two forces.

Proof of Lami's Theorem:

Let's consider a body under the action of three forces F1, F2, and F3 acting at an angle α, β, and γ respectively. According to the problem, the body is in equilibrium.

Step 1: Draw a triangle with the three forces as its sides. The angles opposite to the forces F1, F2, and F3 are α, β, and γ respectively.

Step 2: According to the law of sines in trigonometry, the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

So, we can write:

F1/sinα = F2/sinβ = F3/sinγ

This is the mathematical representation of Lami's theorem.

This theorem is a direct consequence of the equilibrium conditions, which state that the vector sum of all forces acting on a body in equilibrium is zero. It is widely used in statics for solving problems involving three forces.