Imagine an object hanging from the ceiling as shown in the image. Somebody decides to apply a force to the string that the object is attached to. The string bends as a result of that force and forms an angle θ. What is that angle? And what's the new tension of the string?
First, we can label some angles on this free-body diagram. Since the force is applied perpendicularly to the object, the two angles on the right can be deduced to be 90 degrees. The other two angles can then be labeled in terms of θ as shown in the diagram
To solve this problem, we would need to utilize Lami's theorem. It states that when three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. And it looks something like this
If you instantly recognize this from the Sine Rule for triangles, then you are absolutely correct! It is a direct derivation from the Sine Rule. If we represent the forces as lines in a free-body diagram and translate them in such a way that one head touches the tail of another, we will notice that these three forces resultantly form a triangle.
So to solve this, we just have to apply the rule. Very simple.
$$\frac{100N}{\sin (\theta+90)}=\frac{x}{\sin 90}=\frac{20N}{\sin (180-\theta)}$$Apply some trig identities...
$$\frac{100N}{\cos (\theta)}=\frac{x}{1}=\frac{20N}{\sin (\theta)}$$ $$\tan (\theta)=\frac{20}{100}$$ $$x=102N$$ $$\theta=11.3^{\circ}$$As a conclusion, Lami's theorem could be used in any problem involving three forces in equilibrium. When a particular body is under the influence of these three forces to reach equilibrium during which the body remains stationary is best studied using Lami's theorem. Such a simple theorem which originated from Sine Rule is especially ubiquitous and useful in the field of static analysis and structural systems.
