# Question Video: Finding the Triangle of Forces Rule Mathematics

Which of the following represents the triangle of forces rule? [A] πΉβ/π΄πΆ = πΉβ/ππΆ = πΉβ/π΄π [B] πΉβ/ππ΄ = πΉβ/π΄πΆ = πΉβ/ππΆ [C] πΉβ/β π΄ = πΉβ/β πΆ = πΉβ/β π [D] πΉβ/β πΆ = πΉβ/β π = πΉβ/β π΄ [E] πΉβ/sin π΄ = πΉβ/sin πΆ = πΉβ/sin π

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### Video Transcript

Which of the following represents the triangle of forces rule? Is it (A) πΉ sub one over π΄πΆ equals πΉ sub two over ππΆ equals πΉ sub three over π΄π? Option (B) πΉ sub one over ππ΄ equals πΉ sub two over π΄πΆ equals πΉ sub three over ππΆ. Option (C) πΉ sub one over angle π΄ equals πΉ sub two over angle πΆ equals πΉ sub three over angle π. Option (D) πΉ sub one over angle πΆ equals πΉ sub two over angle π, which is equal to πΉ sub three over angle π΄. Or finally, option (E) πΉ sub one over sin π΄ is equal to πΉ sub two over sin πΆ, which is equal to πΉ sub three over sin π.

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 πΉ sub one acts along ππ΄, the force πΉ sub two acts along π΄πΆ, and the force πΉ sub three acts along the third side of our triangle, πΆπ. 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 πΉ sub one proportional to side length π΄πΆ. 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 πΉ sub three acts in the direction πΆπ. However, as this is the side length of the triangle, πΆπ is equal to ππΆ. This confirms that the correct answer is option (B). πΉ sub one over ππ΄ equals πΉ sub two over π΄πΆ equals πΉ sub three over ππΆ 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 Lamiβs 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, πΉ sub one would have to be matched with sin πΆ, πΉ sub two with sin π, and πΉ sub three with sin π΄. Option (E) is therefore not the correct form of Lamiβs 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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