### Video Transcript

In the given figure, particle 𝐴 is in equilibrium under the effect of the forces shown which are in newtons. Find the force 𝐹.

In order to solve this problem, we can 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 also equal to 𝐶 divide by sin 𝛾. As the angle 𝛼 is between forces 𝐵 and 𝐶, the angle 𝛽 is between forces 𝐴 and 𝐶, and the angle 𝛾 is between forces 𝐴 and 𝐵.

In terms of our question, the first step is to calculate the missing angle in the diagram. Well, angles in a circle or at a point add up to 360. Therefore, the missing angle can be calculated by adding 150 to 150 and taking the answer away from 360. 150 plus 150 is 300. And 360 minus 300 is 60 degrees. This means that the missing angle between the two 31-newton forces is 60 degrees.

Substituting our values into Lami’s theorem give us 𝐹 divided by sin 60 is equal to 31 divided by sin 150, which is equal to 31 divided by sin 150. Multiplying both sides of the equation circled by sin 60, gives us that 𝐹 is equal to 31 divided by sin 150 multiplied by sin 60. This gives us a value of 𝐹 of 31 root three.

For the body in the figure to remain in equilibrium, the force 𝐹 is 31 root three newtons, or 53.7 newtons, to one decimal place.