Video Transcript
Resolve a force of 81 newtons into two perpendicular components, 𝐹 one and 𝐹 two, as shown in the figure. Give your answer correct to two decimal places.
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 if we have three forces 𝐴, 𝐵, and 𝐶, then 𝐴 divided by sin 𝛼, the angle between force 𝐵 and force 𝐶, is equal to 𝐵 divided by sin 𝛽, which is equal to 𝐶 divided by sin 𝛾.
Our first step in the figure shown is to work out the missing angle in the diagram. Well, if two forces are perpendicular to each other, then they must meet at 90 degrees. This means that the angle between the force 𝐹 two and the 81-newton force is 36 degrees, as 90 minus 54 is equal to 36.
Applying Lami’s theorem gives us 𝐹 one divided by sin 36, as the 36-degree angle is between the force 𝐹 two and 81. This is equal to 𝐹 two divided by sin 54, as the 54-degree angle is between the force 81 newtons and 𝐹 one. This is also equal to 81 divided by sin 90, as the 90-degree angle is between the forces 𝐹 one and 𝐹 two.
To calculate the force 𝐹 one we will solve 𝐹 one divided by sin 36 is equal to 81 divided by sin 90. Multiplying both sides of this equation by sin 36 gives us that 𝐹 one is equal to 81 divided by sin 90 multiplied by sin 36. This gives us a value for the force 𝐹 one of 47.61 newtons to two decimal places.
In order to calculate the force 𝐹 two, we will solve the equation 𝐹 two divided by sin 54 is equal to 81 divided by sin 90. Multiplying both sides of this equation by sin 54 gives us 𝐹 two is equal to 81 divided by sin 90 multiplied by sin 54. This is equal to 65.32 newtons. Therefore, 𝐹 two is equal to 65.32 newtons to two decimal places.
The two perpendicular components 𝐹 one and 𝐹 two, shown in the figure, are 47.61 newtons and 65.32 newtons.
