Two forces act on a particle. One of the forces has a magnitude of 14 newtons. But the magnitude of the other force is unknown. Given that the resultant force points in the direction of the angle bisector of the two forces, find the unknown magnitude.
The problem can be modelled using the following diagram. Our two forces are 14 newtons and 𝐹. The resultant force is labelled 𝑅. As the resultant force points in the direction of the angle bisector of the two forces, the angle between the 14-newton force and 𝑅 is 𝜃. And the angle between 𝑅 and 𝐹 is also 𝜃.
In order to solve the problem, we can use Lami’s theorem, where each force is proportional to the sine of the angle between the other two forces, such that 𝐴 divided by sin 𝛼 is equal to 𝐵 divided by sin 𝛽, which is also equal to 𝐶 divided by sin 𝛾.
In this case, 14 divided by sin 𝜃, the angle between 𝑅 and 𝐹, is equal to 𝐹 divided by sin 𝜃, the angle between 𝑅 and 14 newtons, which is also equal to 𝑅 divided by sin two 𝜃, the angle between the 14-newton force and 𝐹.
As 14 divided by sin 𝜃 is equal to 𝐹 divided by sin 𝜃, we can say that 14 is equal to 𝐹. Therefore, the magnitude of the unknown force is 14 newtons. We can go one stage further here by saying that if the resultant force bisects any two forces, then the magnitude of those two forces must be equal.