### Video Transcript

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 ๐น holds the body in equilibrium when the measure of the angle between the wall and the string is 30 degrees. Find ๐, the tension in the string, and ๐น, the horizontal force.

In order to solve this problem, we will use Lamiโs 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 ๐ด divided by sin ๐ผ is equal to ๐ต divided by sin ๐ฝ, which is equal to ๐ถ divided by sin ๐พ.

In our example, the angle between the 12-newton force and ๐น is 90 degrees. The angle between the 12-newton force and ๐ is 150 degrees. And finally, the angle between the force ๐น and the tension ๐ is 120 degrees. Substituting these values into Lamiโs theorem gives us ๐น divided by sin 150 is equal to ๐ divided by sin 90, which is equal to 12 divided by sin 120.

Rearranging the two equations circled gives us that ๐น is equal to 12 divided by sin 120 multiplied by sin 150. This gives us a horizontal force ๐น of four root three newtons. Rearranging the two equations now circled gives us ๐ 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 ๐น is equal to four root three newtons and ๐ is equal to eight root three newtons.