Question Video: Equilibrium of a System of Three Forces Acting through a Triangle Mathematics

A body is under the effect of three forces of magnitudes 𝐹₁, 𝐹₂, and 36 newtons, acting in the directions of line segment 𝐴𝐡, line segment 𝐡𝐢, and line segment 𝐴𝐢, respectively, where △𝐴𝐡𝐢 is a triangle such that 𝐴𝐡 = 4 cm, 𝐡𝐢 = 6 cm, and 𝐴𝐢 = 6 cm. Given that the system is in equilibrium, find 𝐹₁ and 𝐹₂.

04:26

Video Transcript

A body is under the effect of three forces of magnitudes 𝐹 sub one, 𝐹 sub two, and 36 newtons, acting in the directions of line segments 𝐴𝐡, 𝐡𝐢, and 𝐴𝐢, respectively, where triangle 𝐴𝐡𝐢 is a triangle such that 𝐴𝐡 equals four centimeters, 𝐡𝐢 equals six centimeters, and 𝐴𝐢 equals six centimeters. Given that the system is in equilibrium, find 𝐹 sub one and 𝐹 sub two.

We know that when three coplanar forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the adjacent sides of a triangle taken in order. So we’re going to represent the three forces given using a triangle. But we’re told that they act in the directions of the various sides of triangle 𝐴𝐡𝐢. So we’ll sketch triangle 𝐴𝐡𝐢 first. Triangle 𝐴𝐡𝐢 looks a little something like this, and we notice that the sides 𝐴𝐢 and 𝐡𝐢 are both six centimeters in length. So it’s actually an isosceles triangle.

We’ll now use this triangle to sketch a triangle of forces representing 𝐹 sub one, 𝐹 sub two, and the 36-newton force. The force with magnitude 𝐹 sub one newtons acts in the direction of line segment 𝐴𝐡. Then the force with magnitude 𝐹 sub two acts in the direction of line segment 𝐡𝐢. Notice that the force with magnitude 𝐹 sub two begins at the terminal point of our previous force. And so we have to begin our third force at the terminal point of 𝐹 sub two. But we were told that this 36-newton force acts in the direction of the line segment 𝐴𝐢, not the line segment 𝐢𝐴. However, since we know that the magnitude of this force is 36 newtons, we can label it as shown. If we were considering the direction of the force, we would need to consider that this would be the negative direction of our original force. But for magnitudes which just represent size, this is absolutely fine.

We’re now ready to compare our triangles. Since each of our forces acts in the same direction as each side in our triangle 𝐴𝐡𝐢, the two triangles must in fact be similar. And so we can say that the magnitudes of each of our forces must be directly proportional to the lengths of the sides in triangle 𝐴𝐡𝐢. So we can find force 𝐹 sub two really easily. We know that the sides 𝐴𝐢 and 𝐡𝐢 are equal in length. So this force and this force must be equal in magnitude. And so 𝐹 sub two must be equal to 36 newtons. And then we have two different ways that we can calculate the magnitude 𝐹 sub one.

One way is to say that the ratio of the line segment 𝐴𝐡 to the ratio of line segment 𝐴𝐢 will be equal to the ratio of the magnitude 𝐹 sub one to the magnitude 36 newtons. In other words, four divided by six will give us the same outcome as 𝐹 sub one divided by 36. And whilst we could simplify the fraction four-sixths, it doesn’t make a lot of sense to do this because we’re going to multiply both sides of this equation by 36. Then we spot that 36 and six have a common factor of six. So 𝐹 sub one will be equal to six times four over one, which is simply equal to 24. And so 𝐹 sub one is 24 newtons. It’s worth noting at this point that we could’ve used scale factor to calculate the value of 𝐹 sub one.

Since the two triangles are similar, we can deduce that one is an enlargement or a dilation of the other. And thus, the scale factor for enlargement would be 36, that’s one of the dimensions on our force triangle, divided by six, the corresponding dimension on triangle 𝐴𝐡𝐢. 36 divided by six is six. And so we can transform any measurement on our triangle 𝐴𝐡𝐢 onto the dimensions of our force triangle by multiplying by six. This means 𝐹 sub one would be equal to four times six, which is once again 24. 𝐹 sub one is 24 newtons and 𝐹 sub two is 36 newtons.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.