In a square 𝐴𝐵𝐶𝐷, 𝑀 is the point of intersection of the two diagonals, 𝐸 is the midpoint of 𝐴𝐵, and 𝐹 is the midpoint of 𝐵𝐶. Three forces of magnitudes 𝐹 one, 𝐹 two, and 41 newtons are acting at 𝑀 in the directions 𝑀𝐸, 𝑀𝐹, and 𝑀𝐷, respectively. Given that the three forces are in equilibrium, find the values of 𝐹 one and 𝐹 two.
Before starting this question, it is worth drawing a diagram. We have a square 𝐴𝐵𝐶𝐷 as shown. We’re told that 𝑀 is the point of intersection of the two diagonals, 𝐸 is the midpoint of 𝐴𝐵, and 𝐹 is the midpoint of 𝐵𝐶. We also know that there are three forces acting at 𝑀. 𝐹 one acts in the direction 𝑀𝐸. 𝐹 two acts in the direction 𝑀𝐹. And finally, we have a 41-newton force acting in the direction 𝑀𝐷. As the three forces are in equilibrium, there are several ways of solving this problem. In this question, we will use Lami’s theorem.
This states that if three forces in equilibrium are acting at a point, in this case 𝐴, 𝐵, and 𝐶, where the angle between forces 𝐵 and 𝐶 is 𝛼, between 𝐴 and 𝐶 is 𝛽, and between 𝐴 and 𝐵 is 𝛾. Then 𝐴 over sin. 𝛼 is equal to 𝐵 over sin 𝛽, which is equal to 𝐶 over sin 𝛾. You may also notice that this is the sine rule formula from trigonometry. In our question, the angle between the force 𝐹 one and 𝐹 two is 90 degrees as they are perpendicular. The two diagonals of a square also meet at right angles. As half of a right angle is 45 degrees, the angle between 𝐹 one and the 41-newton force is 135 degrees. The angle between 𝐹 two and the 41-newton force is also 135 degrees as angles in a circle or at a point sum to 360.
Substituting these values into Lami’s Theorem gives us 41 over sin 90 is equal to 𝐹 one over sin 135 which is equal to 𝐹 two over sin of 135. Let’s consider the first two terms. The sin of 90 degrees is equal to one. This means that 41 is equal to 𝐹 one over or divided by sin of 135. Multiplying both sides of this equation by sin of 135 gives us 𝐹 one is equal to 41 multiplied by sin 135. We could just type this into our calculator. However, we know that sin of 45 degrees is equal to one over root two. This can also be written as root two over two. If the sum of two angles equals 180 degrees, then the sin of both of these angles are equal. Therefore, the sin of 135 is also equal to root two over two.
This means that 𝐹 one is equal to 41 multiplied by root two over two. We can, therefore, say that the force 𝐹 one is equal to 41 root two over two newtons. If we consider the second and third term in Lami’s Theorem, we noticed that the denominators are the same. They’re both sin of 135 degrees. If two fractions are equal and their denominators are equal, this means that the numerators must also be equal. As 𝐹 one is equal to 41 root two over two, then 𝐹 two must also be equal to 41 root two over two. Both of the forces in this case are 41 root two over two newtons.