A crate, seen from above, is pulled with three ropes that have the tensions shown in FIGURE P3.44. Tension is a vector directed along the rope, measured in newtons (abbreviated N). Suppose the three ropes are replaced with a single rope that has exactly the same effect on the crate. What is the tension in this rope? Write your answer in component form using unit vectors.

Video transcript

Welcome back, everyone. We are making observations about three horses that are pulling on a cart that is located at the center of our graph here. And we are told that the three horses are going to be replaced by one horse. And we are tasked with finding what is going to be the resulting tension force on or in the one cable that that one horse is pulling. So to rephrase this, essentially, what we're being asked to do is we're being asked to add up all of the tension forces present component wise. So we'll need an X component and a Y component. So what we'll first start out with is we will start out with our X component here. Now, this is just going to be the summation of the X components of the three tension forces that are already present within our system here. So let's go ahead and work that out. Looking at T one here, we can see that there are no X component or there is no X component to our T one force. So that will just be zero. Now, what about the other two? Well, the way we're going to think about this is that we are gonna draw these dotted lines to the X axis. And what you'll see is that these will be, or T two X, the horizontal component will be our X component of each tension vector. So this will be T three X. And just for the sake of arithmetic, we are going to denote the right direction as the positive X direction and the upwards vertical direction as the positive Y direction. Wonderful. Going back to our equation here. For T two X, we are going to multiply our magnitude of times the cosine of our angle 25. Now, as for T three X, we are going to subtract since it is going in the opposite direction and it will be the magnitude also by the cosine of our angle. And what this gives us for T of X when we plug this into our calculator is 7.79 newtons. Wonderful. So now we are ready to move on to T Y here. So T Y, same thing, we're just gonna have T one Y plus T two Y plus T three Y. And what does this give us? Well, for T one, it is a fully vertical uh vector here. So all of the magnitude will contribute to the Y components making us or giving us 20 newtons here. So then we are going to add our T two Y. So plus the magnitude of our vector this time times the sign of our angle minus since our T three is going in the negative direction here minus the magnitude of our vector times the sign of our angle 45 this time. And what this gives us is that T Y is equal to 12.3 newtons. So now we are ready to construct by component, our final tension here, our X component is given by 7.79 which we multiply by the unit vector I plus our Y component of 12.3, which we multiply by the unit vector J which gives us a final answer of answer choice. A thank you all so much for watching. I hope this video helped. We will see you all in the next one.