The angle between two forces of equal magnitude is 60 degrees and the magnitude of their resultant is 71 root three newtons. What is the magnitude of the forces?
When two forces 𝐹 one and 𝐹 two are acting on a point with resultant 𝑅, where 𝐹 one makes an angle 𝜃 one with 𝑅 and 𝐹 two makes an angle 𝜃 two with 𝑅, we can apply the formula 𝐹 one divided by sin 𝜃 two is equal to 𝐹 two divided by sin 𝜃 one, which is equal to 𝑅 divided by sin 𝜃 one plus 𝜃 two. As the two forces are of equal magnitude, we know that 𝐹 one is equal to 𝐹 two.
As these forces are equal, we can also say that the angle between 𝐹 one and 𝑅 and the angle between 𝐹 two and 𝑅 are also equal: 𝜃 one equals 𝜃 two. As the angle between the two forces was 60 degrees, we can see that the angle between 𝐹 and 𝑅 is 30 degrees. This gives us the equation 𝐹 divided by sin 30 is equal to 71 root three divided by sin 60.
Rearranging this equation gives the force 𝐹 is equal to 71 root three divided by sin 60 multiplied by sin 30. This is equal to 71 newtons. Therefore, the magnitude of the forces is 71 newtons.