What is proof of Lami's theorem?

Answer 1

As you probably know, Lami's Theorem only applies to objects in equilibrium, with 3 coplanar (in the same plane) concurrent (intersecting at the same point) forces acting on it. It works because you add vectors together from tip to tail and also taking direction into account, and because the net force of an object in equilibrium is zero.

Let's look at an object for which Lami's Theorem works.

Now, let's add all these forces together, tip to tail.

The force vectors have to do this (form a closed shape) because the object is in equilibrium, and this makes the net force zero. When the net force is zero, the forces should cancel each other out entirely, meaning that adding the vectors will result in zero.

(If we added the force vectors of an object NOT in equilibrium, we would obtain a shape that:

· is not a proper closed shape, i.e. you add the vectors and they form 1. a wonky line, or 2. a weird triangle thingy where you haven't used the entirety of a vector for the shape.

1.

2.

· is some other shape.

This would indicate a net force being present.)

Let's take our added-up forces shape and add some details to it. (By the way, this shape is called a forces triangle.)

All I did was lengthen the force lines in the direction of the vector.

Now:

I just used the original diagram, and found out which angles are between which vectors, and inserted them here into the diagram.

Then, the inside angles must look like:

Now, what's the sine rule again?

For sides a,b,and c and included angles A,B and C:

Let's do it for our forces triangle!

but we know that sin (180-α/β/θ)=sin (α/β/θ), so

And that's Lami's Theorem!

Answer 2

Lami's theorem states that if three concurrent forces act on a body keeping it in Equilibrium, then each force is proportional to the sine of the angle between the other two forces.

Let P, Q and R be the three forces acting at a point O.

Since OP, OQ and OR are vectors, they can be arranged "nose to tail". Then since the forces are in equilibrium, they form a closed triangle.

Applying the sine rule to the triangle, P/sin[pi - angle(QOR)] = Q/sin[pi- angle(ROP)] = R/sin[pi- angle(POQ)]

Then, since sin(pi - x) = sin(x)

P/sin[angle(QOR)] = Q/sin[angle(ROP)] = R/sin[angle(POQ)]