The angle between forces 𝐹 sub one
and 𝐹 sub two is 112 degrees, and the measure of the angle between their resultant
and 𝐹 sub two is 56 degrees. If the magnitude of 𝐹 sub one is
28 newtons, what is the magnitude of 𝐹 sub two?
Let’s begin by sketching the two
forces with an angle of 112 degrees between them. We are also told in the question
that the angle between the resultant force and 𝐹 sub two is 56 degrees. 112 minus 56 is equal to 56. Therefore, the angle between the
resultant and 𝐹 sub one is also 56 degrees.
Using our knowledge of vector
forces, we can create a parallelogram as shown. This can be split into two
congruent triangles. Since 𝐹 sub one is equal to 28
newtons, we can use the sine rule to calculate 𝐹 sub two. The sine rule states that 𝑎 over
sin 𝐴 is equal to 𝑏 over sin 𝐵, where the angles capital 𝐴 and capital 𝐵 are
opposite the side lengths 𝑎 and 𝑏.
Substituting in our values, we have
28 over the sin of 56 degrees is equal to 𝐹 sub two over the sin of 56 degrees. Multiplying through by the sin of
56 degrees, we get 𝐹 sub two is equal to 28. The magnitude of the force 𝐹 sub
two is equal to 28 newtons.
We notice that this is the same
value as 𝐹 sub one, which leads us to a general rule. If the resultant force bisects two
forces, then the two forces will have the same magnitude. In this question, as the angle
between the resultant force and 𝐹 sub one was 56 degrees and the angle between the
resultant and 𝐹 sub two was also 56 degrees, then 𝐹 sub two and 𝐹 sub one must
have the same magnitudes.