A force of magnitude 96 newtons acts vertically downward. It is resolved into two components as shown in the diagram. Find the magnitude of 𝐹 one and 𝐹 two, giving values to two decimal places.
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.
𝐴 divided by sin 𝛼 is equal to 𝐵 divided by sin 𝛽 is equal to 𝐶 divided by sin 𝛾. This is because angle 𝛼 is between forces 𝐵 and 𝐶. Angle 𝛽 is between forces 𝐴 and 𝐶. And finally, angle 𝛾 is between forces 𝐴 and 𝐵.
In our example, the angle between force 𝐹 one and 96 newtons is 90 degrees. The angle between 𝐹 two and 96 newtons is 42 degrees. And the angle between forces 𝐹 one and 𝐹 two is 132 degrees. This means that 𝐹 one divided by sin 42 is equal to 𝐹 two divided by sin 90, which is equal to 96 divided by sin 132.
If we consider the forces 𝐹 one and 96 newtons, we can rearrange the equation so that 𝐹 one is equal to 96 divided by sin 132 multiplied by sin of 42. This gives us an answer for 𝐹 one of 86.44 newtons. When we consider the two forces 𝐹 two and 96 newtons, we can rearrange this equation to give us 𝐹 two is equal to 96 divided by sin 132 multiplied by sin 90. This gives us a value of 𝐹 two of 129.18 newtons.
Using Lami’s theorem, we’ve proved that the force 𝐹 one is equal to 86.44 newtons and 𝐹 two is equal to 129.18 newtons.