### Video Transcript

๐ด๐ต๐ถ๐ท is a square, where the five forces, measured in newtons, are acting on it as shown in the figure. If the system of forces is equivalent to a couple, determine ๐น sub one and ๐น sub two.

We see in the diagram that there are five total forces โ one of 20 newtons, one of 13, one of nine square root of two, ๐น one, and ๐น two โ acting on the corners of the square. In particular, the forces originate from two of the corners of the square: corner ๐ด, where three of the forces originate, and corner ๐ถ, where the other two do.

Weโre told that if we consider all five of the forces together that theyโre equivalent to a force couple. And given that information, we want to solve for ๐น sub one and ๐น sub two. To begin solving for ๐น sub one and ๐น sub two, we can recall the definition of a force couple, that itโs a pair of parallel forces with equal magnitude and opposite direction, which do not lie on the same line of action.

This means that the overall forces that originate at point ๐ด and point ๐ถ should be parallel. They should have equal magnitude and opposite direction. Letโs consider what those forces are based on the information given.

At each of these two corners, weโll endeavor to solve for the vertical as well as the horizontal components of the net force from each corner. Starting with corner ๐ด, we can see as we consider the horizontal component of this force that it has two contributors. Thereโs the force of 13 newtons acting completely horizontally. And thereโs the horizontal component of the force of nine times the square root of two newtons.

Since the direction of that force bisects a 90-degree angle, we know that the horizontal component is equal to the magnitude of the force overall, nine times the square root of two, multiplied by the cos of 45 degrees. And since the cos of 45 degrees is equal to the square root of two over two, our expression simplifies to 13 newtons plus nine newtons, or 22 newtons. Thatโs the horizontal component of the force originating at point ๐ด.

When we consider the vertical component of that force, we see that it consists of the vertical component of nine times the square root of two, which is nine root two times the sin of 45 degrees, plus the unknown force ๐น sub two.

Just as with the horizontal component, we can write the vertical component of the force bisecting our 90-degree angle as nine times the square root of two times the square root of two over two, or simply nine newtons. So the vertical component of the total force acting from point ๐ด is nine newtons plus ๐น sub two.

Now we move on to consider the forces originating at corner ๐ถ. When we look at the vertical component of this force, we see that itโs equal seemingly to 20 newtons. And the horizontal component is equal to ๐น sub one. Because these two net corner forces form a force couple, we can generate two equations from the information we figured out so far.

First, we know that the horizontal components of the two forces, 22 newtons in the one case and ๐น sub one in the other, are equal. That is, ๐น sub one is equal to 22 newtons. And further, the vertical components of these two forces, nine newtons plus ๐น sub two and 20 newtons, are also equal to one another. This implies that ๐น sub two equals 20 newtons minus nine newtons, or 11 newtons.

So weโve solved for ๐น sub one and ๐น sub two, the force magnitudes which make the overall forces in this diagram equal a force couple.