Question Video: The Resultant of Two Nonperpendicular Forces Mathematics

The angle between forces 𝐹₁ and 𝐹₂ is 112Β°, and the measure of the angle between their resultant and 𝐹₂ is 56Β°. If the magnitude of 𝐹₁ is 28 N, what is the magnitude of 𝐹₂?

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Video Transcript

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 π‘Ž over sin 𝐴 is equal to 𝑏 over sin 𝐡, where the angles capital 𝐴 and capital 𝐡 are opposite the side lengths π‘Ž 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.

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