Question Video: Analysis of the Resolved Components of a Force Mathematics

The angle between two forces 𝑎₁ and 𝑎₂, is 75°. Their resultant is 2,900 N and makes an angle of 45° with 𝑎₁. Find the forces 𝑎₁ and 𝑎₂. Give your answers to two decimal places.

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Video Transcript

The angle between two forces, 𝑎 one and 𝑎 two, is 75 degrees. Their resultant is 2,900 newtons and makes an angle of 45 degrees with 𝑎 one. Find the forces 𝑎 one and 𝑎 two. Give your answers to two decimal places.

Before we do anything, we’re going to begin by drawing a force diagram. Here are our two forces 𝑎 one and 𝑎 two, which meet at an angle of 75 degrees. Then, we have this resultant force of 2,900 newtons that makes an angle of 45 degrees with 𝑎 one. Remember, the resultant force is just the single force that we get by combining a system of forces, so here 𝑎 one and 𝑎 two. Our job is to find the forces 𝑎 sub one and 𝑎 sub two. And so we’re going to add a couple of things to our diagram.

Firstly, we add in lines parallel to those representing the forces 𝑎 one and 𝑎 two. This gives us a parallelogram. And so we know that not only are the opposite sides parallel, but they’re also equal in length. And that’s going to be useful later down the line. Then we use the fact that alternate angles are equal. And this angle must be 45 degrees. Then, by subtracting 45 from 75, we get two angles of 30 degrees. The third angle is found by subtracting 45 and 30 from 180 degrees, since there are 180 degrees in a triangle. And so we see that the third angle in both our triangles is 105 degrees.

Let’s separate out our triangles. And we see that we’re dealing with a pair of identical non-right-angled triangles. We have a length of 2,900 in both triangles. And then we have 𝑎 sub one here. And since we said that opposite sides in a parallelogram are parallel and equal in length, we have 𝑎 sub two over here.

Since we know the measure of all the angles in our triangle and one of its lengths, we can use the law of sines to find the other two. Let’s label our triangle as shown. And to begin with, we’ll work out the value of 𝑎 sub one. This means we want 𝑎 over sin 𝐴 equals 𝑐 over sin 𝐶. Substituting everything we know about our triangle into this formula gives us 𝑎 sub one over sin of 30 degrees equals 2,900 over sin of 105. We’re going to multiply both sides by sin of 30 to find the value of 𝑎 sub one. And so 𝑎 sub one is 2,900 over sin of 105 times sin of 30, which correct to two decimal places is 1,501.15. So 𝑎 sub one is 1,501.15 newtons.

Let’s clear some space and perform the same process to find 𝑎 sub two. This time, we’re going to use 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶. Substituting into the formula, and we see that we can solve for 𝑎 sub two by multiplying both sides by sin of 45. And 𝑎 sub two then is 2,900 over sin of 105 times sin of 45. And that, correct to two decimal places, is 2,122.95. 𝑎 sub one is 1,501.15 newtons, and 𝑎 sub two is 2,122.95 newtons.