Find the angle between two vector forces whose magnitudes are F and 3F and the magnitude of whose resultant is 3F.

Question

Find the angle between two vector forces whose magnitudes are F and 3F and the magnitude of whose resultant is 3F.

Answer 1

Since we're only dealing vector magnitudes and know the magnitudes of both the two vectors and of the resultant, we can treat this like a triangle. Because we know that we have two sides of 3F and one side of F, we are dealing with an isosceles triangle.

If we bisect the triangle from the vertex between the two 3F sides to the midpoint of the F side, we get two right-angle triangles, which makes our job easier.

Taking a look at one of these right-angle triangles, we have a side of 3F, a side of 0.5F (since we split the F side into two equal parts), and a side of unknown length. We could calculate this unknown side, but we don't need to, since we can take the cosine of the angle between the 3F and 0.5F sides to solve this problem.

Let's do that:

cos(Θ) = 0.5F/3F = 0.5/3 = 1/6

Θ = cos^-1(1/6)

Θ = 80.41°