Hello and welcome to Chapter 16. Problem number 16 in principles of physics. So this problem tells us that we have two pieces of wood to form a junction here with an elbow at origin. Oh, on they follow our coordinates. Positive. Why? With R l positive expert H. It also tells us that we're gonna have ah, thermal expansion of this conjunction here when water pipe is turned on on giving us a temperature change a swell. So I have our thermal expansion equation here. Delta l is Alfa Times. L not times Delta t I as well have the thermal expansion coefficient for copper. That's the material that would be expanding. Hearing problem eso First things first. We need Teoh. We're gonna be first calculating our change in both the horizontal and vertical parts. That's gonna be the total as its of the problem asks us, we need to find that total displacement here, so we need to calculate our our change and lengths. So we first need to get that delta T temperature. Just get that is a number. And then we can, um, find our individual delta elves for both vertical and horizontal components. So our Delta T in the beginning is just gonna be that 45 point are 46.5 minus 18. So we'll have about 28.5 degrees of change here. Now that we have that we can pretty much just plug in our individual values for or l knots And then are coefficient of thermal expansion for copper as well That Delta team. So the delta l for our vertical component, it's gonna be 17 times in the line of sex times at 0.28 Remember, we also have to do We'll have to convert our our lengths. And commuters, I have that up here. They give us, give it to us and centimeters Regardless, we have 17. Excuse me, 17 Thompson of my six times 60.28 times at 28.5 and we'll have a verdict component. A change of 1.35665 cent alliance for and the very same process for our horizontal component. You just have ah, 1.34 as are all not in this equation here. So we have both of our components here. But again, these air just the Delta Els. I'm gonna go back up toward diagram here. This is just the Delta L, um, And for both of these individually, it asks us the total displacement and direction of this elbow at origin. Oh, here. So what we're gonna actually get to do next is used the distance formula to find that magnitude. Eso, if you recall distance formula is gonna be one component of the of our of our direction. Change squared, plus the other components square. Take the square root of that. So, um, we're gonna have d when we plug in our numbers. Here. Be about 6.63 times 10 to the minus, four meters on. Just again. Reminder. Um, this is Delta L l 1.3566 times in the lines. Four squared plus 6.4923% of the minus four squared and then take the square root of that. So that's our distance here on now. In order to find the direction we're gonna look at Delta l, uh, of the vertical and horizontal components as part of a triangle is well, and then we're gonna use trigonometry. I chose this data here. You can choose the angle up here is Well, uh, he or again, it doesn't really matter which angle uses lines you have give that the sense of direction here. So I have, um, using, you know, pretty standard trigonometry rules we have there's equal toward inverse tangent are arc tangent of the opposite toe. Uh, if you remember, so Cotto of the opposite component l vertical over our horizontal are adjacent component l horizontal. So inverse tangent of delta L for the vertical and then divided by Delta off of the horizontal. And again, when we plug those numbers in, we're going to get about 78.20 degrees.