# Question Video: Finding the Triangle of Forces Rule Mathematics

Which of the following represents the triangle of forces rule? [A] 𝐹₁/𝐴𝐶 = 𝐹₂/𝑂𝐶 = 𝐹₃/𝐴𝑂 [B] 𝐹₁/𝑂𝐴 = 𝐹₂/𝐴𝐶 = 𝐹₃/𝑂𝐶 [C] 𝐹₁/∠𝐴 = 𝐹₂/∠𝐶 = 𝐹₃/∠𝑂 [D] 𝐹₁/∠𝐶 = 𝐹₂/∠𝑂 = 𝐹₃/∠𝐴 [E] 𝐹₁/sin 𝐴 = 𝐹₂/sin 𝐶 = 𝐹₃/sin 𝑂

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### Video Transcript

Which of the following represents the triangle of forces rule? Is it (A) 𝐹 sub one over 𝐴𝐶 equals 𝐹 sub two over 𝑂𝐶 equals 𝐹 sub three over 𝐴𝑂? Option (B) 𝐹 sub one over 𝑂𝐴 equals 𝐹 sub two over 𝐴𝐶 equals 𝐹 sub three over 𝑂𝐶. Option (C) 𝐹 sub one over angle 𝐴 equals 𝐹 sub two over angle 𝐶 equals 𝐹 sub three over angle 𝑂. Option (D) 𝐹 sub one over angle 𝐶 equals 𝐹 sub two over angle 𝑂, which is equal to 𝐹 sub three over angle 𝐴. Or finally, option (E) 𝐹 sub one over sin 𝐴 is equal to 𝐹 sub two over sin 𝐶, which is equal to 𝐹 sub three over sin 𝑂.

The triangle law of forces states that if a body is in equilibrium under the action of three forces at a point, then the three forces can be represented by the three sides of a triangle in order. This can be seen in our diagram, where the force 𝐹 sub one acts along 𝑂𝐴, the force 𝐹 sub two acts along 𝐴𝐶, and the force 𝐹 sub three acts along the third side of our triangle, 𝐶𝑂. The magnitude of these forces will therefore be proportional to the length of their corresponding sides.

Options (A) and (B) both have the three forces written as proportions to the lengths of the three sides. Option (A) has 𝐹 sub one proportional to side length 𝐴𝐶. However, this is not its corresponding side. Option (B), on the other hand, has the correct forces matched to the correct sides. It is worth noting that the force 𝐹 sub three acts in the direction 𝐶𝑂. However, as this is the side length of the triangle, 𝐶𝑂 is equal to 𝑂𝐶. This confirms that the correct answer is option (B). 𝐹 sub one over 𝑂𝐴 equals 𝐹 sub two over 𝐴𝐶 equals 𝐹 sub three over 𝑂𝐶 represents the triangle of forces rule.

Options (C) and (D) match the three forces to angles in the triangle. Whilst option (D) has matched the forces to the angles opposite them, this is not the triangle of forces rule. The forces are not proportional to the angles in the triangle. Option (E) looks like it could be Lami’s theorem. This states that when three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. For this to be true, 𝐹 sub one would have to be matched with sin 𝐶, 𝐹 sub two with sin 𝑂, and 𝐹 sub three with sin 𝐴. Option (E) is therefore not the correct form of Lami’s theorem. However, even if this was the case, this would not be the correct answer for this question as we were looking for the equation that represents the triangle of forces rule.

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