A body weighing 72 newtons is placed on a plane that is inclined at 45 degrees to the horizontal. Resolve its weight into two components 𝐹 one and 𝐹 two, where 𝐹 one is the component in the direction of the plane and 𝐹 two is the component normal to the plane.
The forces 𝐹 one and 𝐹 two are perpendicular to each other. Therefore, the angle between them is 90 degrees. The angle between the force 𝐹 one and the weight 72 newtons is 45 degrees and the angle between force 𝐹 two and 72 newtons is also 45 degrees. This question can be solved using Lami’s theorem, which 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 which is equal to 𝐶 divide by sin 𝛾, where the angle 𝛼 is between the forces 𝐵 and 𝐶. 𝛽 is the angle between forces 𝐴 and 𝐶 and 𝛾 is the angle between forces 𝐴 and 𝐵. Substituting our values into Lami’s theorem gives us 𝐹 one divided by sin 45 is equal to 𝐹 two divided by sin 45, which is equal to 72 divided by sin 90.
We can rearrange the equation circled to give us 𝐹 one is equal to 72 divided by sin 90 multiplied by sin 45. This gives us a value of 𝐹 one of 36 root two or 50.91 to two decimal places. This means that the force 𝐹 one in the direction of the plane is 36 root two newtons.
We can solve to calculate 𝐹 two in a similar way. 𝐹 two is equal to 72 divided by sin 90 multiplied by sin 45. This means that 𝐹 two is also equal to 36 root two. The component of the force normal to the plane is 36 root two newtons.
We can go one step further here by saying that two perpendicular forces 𝐹 one and 𝐹 two will be equal to each other when the body is inclined at 45 degrees to the horizontal. This is because 45 degrees bisects the perpendicular forces.