A body weighing 12 newtons is attached to one end of a light, inextensible string. The other end of the string is fixed to a vertical wall. A horizontal force 𝐹 holds the body in equilibrium when the measure of the angle between the wall and the string is 30 degrees. Find 𝑇, the tension in the string, and 𝐹, the horizontal force.
In order to solve this problem, we will use Lami’s theorem. This states that if 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. This means that 𝐴 divided by sin 𝛼 is equal to 𝐵 divided by sin 𝛽, which is equal to 𝐶 divided by sin 𝛾.
In our example, the angle between the 12-newton force and 𝐹 is 90 degrees. The angle between the 12-newton force and 𝑇 is 150 degrees. And finally, the angle between the force 𝐹 and the tension 𝑇 is 120 degrees. Substituting these values into Lami’s theorem gives us 𝐹 divided by sin 150 is equal to 𝑇 divided by sin 90, which is equal to 12 divided by sin 120.
Rearranging the two equations circled gives us that 𝐹 is equal to 12 divided by sin 120 multiplied by sin 150. This gives us a horizontal force 𝐹 of four root three newtons. Rearranging the two equations now circled gives us 𝑇 is equal to 12 divided by sin 120 multiplied by sin 90. This is equal to eight root three.
Therefore, the tension in the string is eight root three newtons. The body remains in equilibrium when 𝐹 is equal to four root three newtons and 𝑇 is equal to eight root three newtons.