 So in section 9.1 of our lecture series, we introduced the idea of vectors and their geometric or physical representation, these arrows. And we talked about the magnitude, the direction, and the components of a given vector arrow there. And we were able to use that to help us understand some physical ideas about direction and speed of moving objects. In section 9.2, I want to continue these ideas of vector applications, focusing on things from physics, mostly on directions and force vectors of some kind, and show you really how the parallelogram rule when it comes to adding vectors naturally leads itself to trigonometric type problems. And so in this video and the next one, I want to focus on a specific problem called static equilibrium. So when an object is stationary, we say that it's in static equilibrium. That is, it's stationary, it's not moving. Now, if it's not moving, that does not mean that there's no forces acting on it. Most likely, I mean I could be wrong, but most likely you yourself as you are watching this video are in a state of static equilibrium. You're probably sitting, your computer or device you're watching on is probably also not moving relative to the Earth, of course, which is moving. But we'll get away from that issue right there, right? It's not moving, but that means there doesn't mean there's no forces acting on it. Like if you take me for example, I am not moving at the moment. I am sitting in a chair, which we'll just think of it like this right here. Here's my chair. But there are forces acting upon it right now, right? Gravity is pulling this chair towards the center of the Earth. On the other hand, there is a force exerted by the ground pushing back against gravity and these forces are equal and opposite in their direction, which is why my chair or myself is not accelerating towards the center of the Earth. If you don't believe in normal force, ask your big brother to sit on your head sometime and then you will feel his weight, but you also feel the force of the floor pushing back as you feel you're still being crushed right there, right? And so static equilibrium means that the sum of all the forces acting on an object is the zero vector. There's no motion because the net sum is zero. And so that's a very important thing for us here because when it comes to vectors, when you add two vectors together, right, you put the tail to the head, something like this. If the force is zero, that would mean that in the end, when you connect all of these forces together, just drawing some hypothetical examples here, when you put all of the forces together, head to tail, head to tail, head to tail, static equilibrium means you form a polygon. That is, you eventually stop where you started. The net force is zero. And so when you have three vectors in play here, a polygon is a triangle. Ooh, trigonometry seems like it might be relevant for such a situation. Let's consider such an example. So Danny is a five-year-old little boy and he weighs 42 pounds. So that's his weight, which is the force due to gravity. He's sitting on a swing and his big sister, Stacy, she pulls him up, right? Kids like to be suspended in the air on the swing in a suspenseful moment like, what's going to happen when I fall? Well, of course, you'll start swinging, but you know, you worry you're going to fall to your doom. Clearly, it's not going to happen. But Stacy, she's holding Danny back on his swing in such a way that it forms a 30-degree angle and she's holding him right there. So this little diagram is to try to show you what's going on right here. So this right here represents Danny. He's at the middle of all of this. His weight is pushing down on him at 42 degrees. But like I said, Stacy's pulling him and suspending him in the air. But then there's also a chain or a string or a rope of some kind, depends on the type of swing there, right? But there's some type of rope that's suspended to the swing and it has a tension. So the tension is how much force is being pulled on our rope or chain, what have you. And it forms an angle so that with respect to the vertical here, this tension is 30 degrees like so. So we have the rope of the swing. We have the sister pulling, holding the boy up and then he has his weight. So there are three vectors. These three forces are acting on Danny all at the same time. So gravity pushes him down. Stacy's pulling him to the side, but the rope is also pulling him back towards the swing set. But he's not moving. He's in static equilibrium, which means that the combined forces of his weight, Stacy and the tension, they add up to be zero. They have static equilibrium. It adds up to be zero in that situation. Now in this format, it might not be exactly sure what we're going to do with this thing here. But what we want to do is we want to find out the tension of the ropes of the swing. So how hard is the rope pulling Danny? And also how hard is Stacy pulling Danny, right? What are the magnitudes of these vectors? Now the weight vector is pretty easy. That was given to us. Danny weighs 42 pounds. Now pounds is a unit of force. It measures how hard something is being pushed. So pounds represents how hard is gravity pulling on things. So the weight of Danny is 42 degrees and gravity always points downward. So we know it's going to go in a downward position. We were told that Stacy pulls him in a horizontal direction. So that forms a right angle here. And then you also give this mention of the 30 degrees, which we see in the diagram is representing this angle right here is 30 degrees. So because we're in static equilibrium, what we can do is we can move the arrows around thus to form a polygon, which in the case there's three vectors, here's going to be a triangle. So what happens if I were to move the force vector up like so? We have this weight vector up like so. We end up with something like the following intermediate diagram. So we have our weight vector. We have Stacy pulling like so. And then we have the tension coming off like so. And then the next thing is what if we move the tension over here? Again, what we're trying to do is we're trying to move it so that everything goes from head to tail. So tail to head, tail to head, tail to head like so. We can rearrange these vectors in the following picture like so, so that we then get this diagram for static equilibrium. You should be able to form a polygon right here. Now the angle between S and W should still be a right angle. What about this 30 degrees? How do I conclude this is 30 degrees right here? Well, notice that these lines right here are one parallel to each other. So the weight is a vertical vector pointing downwards. And the tension forms a 30 degree angle with respect to the vertical as we see illustrated right here. So these two lines as they're both vertical lines are parallel to each other. But then the tension vector, if we extend that line, it forms a transversal of two parallel lines. And so this angle right here, its corresponding angle would be this angle right here. And so by the alternate to your angle theorem, those angles will actually be congruent to each other. And so that's why we are able to conclude that this is a 30 degree angle inside of this triangle. And so now let's draw our triangle with our new light. We have this triangle, it's a right triangle. It's actually a 30, 60, 90 triangle if we want to be more precise. We know this side is 42, like so, 30, 60, 90 triangle, oops, 60 degrees. And we want to figure out the missing pieces here. So the hypotenuse is going to be the tension. And then this right here, the magnitude would be how hard Stacy's pulling this thing. Well, since it's a 30, 60, 90 triangle, we can dramatically simplify the trigonometry here. The hypotenuse will be twice the short leg, right? Which we don't know what that is yet, right? We know the leg associated to 60 degrees. So to get 30 degrees, we're going to take the 42 and divide it by the square root of 3, like so. Then to get the tension, we would double that. And so we end up with 84 over the square root of 3, like so. Which 84 over the square root of 3, that's going to be approximately 48.5 pounds. So the rope is holding what feels like about 50 pounds. So that's heavier than Danny, right? Stacy, on the other hand, she's holding him up. You take 42 over the square root of 3, you'll end up with 24.2 pounds, like so. So from Stacy's perspective, she's not holding the whole weight of Danny. Because honestly, the rope is doing a lot of the work there. It's doing more work than it was unnecessary. That's because the rope is picking up the slack from Stacy as well. And so we see that the rope, it has a tension of 48.5 pounds. And Stacy has, she's holding up Danny with a force of 24.2 pounds right there. And so this little example of static equilibrium shows you how when one works with force vectors, it naturally leads itself to trigonometric settings. Now this question was easy compared to what you could see for two reasons. One, the relationship here that Stacy's holding, her force was horizontal to the weight. You have this right angle. This forms a right triangle, which makes the trigonometry so much easier with right triangles. You just have your usual Soca Toa. That's one thing that made it easy. Plus also, since it was a 30, 60, 90 triangle, that also simplified the trigonometry dramatically. But of course, with the use of a scientific calculator, if we had something like a 20 degree angle, that wouldn't have made much of a difference. We just take sine and cosine of 20 degrees. We can handle such a thing, right? And but what happens when you lose the right angle here? Because we don't always expect these forces to be acting perpendicular to each other. So as we continue section 9.2 in the next lecture, lecture 29, that is, we'll look at examples of forces that are acting in static equilibrium, but don't necessarily form right angles. Thus, complicated trigonometry, but not beyond what we've already learned in this lecture series.