# Question Video: Finding the Magnitude of Two Forces in a Coplanar Couple Mathematics

π΄π΅πΆπ· is a square. The forces πΉ, 2N, πΉ, and 2N act along the directions π΄π΅, πΆπ΅, πΆπ·, and π΄π·, respectively. Find the value of πΉ for the system to be in an equilibrium state.

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### Video Transcript

π΄π΅πΆπ· is a square. The forces πΉ, two newtons, πΉ, and two newtons act along the directions π΄π΅, πΆπ΅, πΆπ·, and π΄π·, respectively. Find the value of πΉ for the system to be in an equilibrium state.

We begin by sketching the square π΄π΅πΆπ· as shown. There are four forces πΉ, two newtons, πΉ, and two newtons acting along the four sides of the square. These forces form two couples, where a couple is a pair of parallel, but not coincident, forces of equal magnitudes and opposite directions. We have been asked to find the value of πΉ for the system to be in an equilibrium state. And we know for this to be true, the sum of the moments must equal zero. And the moment of a couple is equal to the magnitude of the force multiplied by the distance between the lines of action.

In this question, both couples have forces acting at points π΄ and πΆ. This means that the distance between the lines of action will be the same for both couples. Convention dictates that moments acting in a counterclockwise direction are positive. In this question, the two-newton couple is acting in a counterclockwise direction. So the moment of this couple will be equal to two multiplied by π.

As the couple of force πΉ is acting in a clockwise direction, this will have a negative moment equal to negative πΉ multiplied by π. We know that the sum of these two moments equals zero. Simplifying the left-hand side, we have two π minus πΉπ is equal to zero. Since the distance π cannot equal zero, we can divide through by this. We can then add πΉ to both sides such that πΉ is equal to two. And we can therefore conclude that the value of πΉ for the system to be in an equilibrium state is two newtons.