 It has to be kind of a rough summary of the entire unit. I will try, OK? Some key ideas. When we talk about, you know what? I'm not going to try and tell you. I just have to live with my messy handwriting. When we talk about circular force, it's mass times circular acceleration. There is, I don't know if you noticed, there's no circular force equation on your formula sheet. What there is is two equations for centripetal or circular acceleration. Either of those gives you the same answer. I would use this one if they gave me the speed or asked me for the speed. I would use this one if I knew the period or they wanted me to find the period. Some key ideas. We said the direction. So if you're moving in a circle, let's try drawing a circle, Mr. Dewick. That's no good. Let's try drawing a circle, Mr. Dewick. Is that a circle for you guys? On my screen, it's not, but the projector distorts everything. So if you're moving in a circle, there's three directions. Let's say, let's put the object right here. And let's pretend you're moving in this direction around a circle. Something is spinning, let's say. What direction is the velocity in A, B, or C? Desmond, can you contact Mr. Nixon, please, in the office? Desmond, contact Mr. Nixon? A. Fancy word, tangent to radius. What direction is the acceleration in? A, B, or C? B. What direction is the force in? A, B, or C? Because force is mass times acceleration, they've got to have the same vector. And the fancy word for this was centripetal. And the phrase that we used was we said it was center seeking. By the way, let's suppose there was a string right here. If I cut the string, which way would the object move off? Direction A or B or C? A, same direction as the velocity. Because as soon as I cut the string, that inwards force is gone, so now it's going to move Newton's first straight law, straight line constant speed. I don't think this is on your formula sheet. Is circular velocity on your formula sheet? You have it right there, Pat, can you check? We said that in case you ever need to figure out the velocity when you're moving in a circle, this shows up once in a while, we said this. Velocity is distance over time, assuming you're going at a steady speed. And when you're moving in a circle at a steady speed, your acceleration is inwards, but your speed is not changing. So this does work. What's the distance around a circle? We call it the circumference. What's the equation for the circumference? And what variable have we used when we go around once? Period. The circular velocity, if they ever ask you how fast is an object traveling in a circle, it's 2 pi r times the time that I divided by the period. That's distance over time, it's meters per second. That's the tangent to radius velocity. The last key idea that we said, FC is net force. It never appears body, Mr. Deweyke, diagram. It's going to be winner minus loser equals, if you're moving in a circle, FC. It never appears on a free body diagram. That's what I mean when I said it never appears on a free body diagram. I'm going to do some examples, can you hang on? But it's never going to appear ever. You'll never see me draw FC on a diagram. Ever. Ready? Let's look at some examples. The first thing we looked at was horizontal circles. I'll type just that once. Nope, too far. Can you bear with me while I find the horizontal circle or two? So here's one right here. Athlete runs at a constant speed around a circle of radius 5 in 12 seconds. What are the athletes speed and acceleration? OK. Let's do acceleration first. Acceleration is v squared over r. Do I know how fast this athlete is traveling? Do I know how fast this athlete is traveling? Nope. By the way, this is number two from the review. I'll pick some from the review if you want to. You can follow along or you can just watch. So I'll use the other circular acceleration. What's the other circular acceleration equation? 4 pi squared r over t squared. Do I know the radius? Yep. Do I know the period? Yep. Good. It's going to be 4 pi squared times 5 over 12 squared. What's his circular acceleration? Someone crunch that for me, please? I get 1.4 meters per second squared. So I would right away do that. Yes, no. Yes, I already know the answer is b or d. Now we need to find his speed. There's two ways we could do this. We could find his speed from there. 1.37 equals v squared over 5 times by 5 square root. And I might do that since I have it on my calculator already. I'm going to go times by 5 square root. I get 2.6. Or the other way, remember we said velocity is 2 pi r over the period. It's circumference over time. That'll also work if I put the data back up there. 2 times pi times 5 divided by 12. And I also get 2.6179999. Correct answer, d. Now that wasn't actually doing any free body stuff. Let's find a horizontal circle. We've got to do some free body stuff. I know of a few questions I'm thinking about there. I'm going to go view, page display, single page continuous though. That's vertical, that's vertical, that's vertical. A car travels 25 meters per second along a horizontal curve of radius 450. What's the minimum coefficient or friction necessary between its tires and the road in order for the car not to skid? So I'm going to do a picture. Here's the car looking right at us. So there's the front. There's the headlights. There's the roof. There's the tires. He's coming right towards us. And he's going around a circle in that direction there. So here's the road. Is that OK? That's my high tech, very artistic free body diagram as good as you're going to get. What are the forces acting on this car? Get the obvious one. MG down. Is the car falling into the pavement? Then there must be a normal force pushing it up. How is this car moving? What path is it tracing out? A circle. Where must the net force be towards the center? There has to be a force that way, but it is not going to appear on my diagram as FC. I need to figure out what force that there is. What force is pushing this car in a circle? But do you see an FC anywhere in my diagram? That's what I was getting at. Horizontally, winner, no loser, equals FC, because we're moving in a circle. Friction is what times what? I don't know the normal force. Let's see here. UMG equals M. I don't forget the M. Do I want to use the one with the period in it, or do I want to use the one with the speed in it? What if they give me this question for acceleration? Sorry. Now I started to panic for a second there. It hesitated because I was going, they didn't tell me the mass of the car. Wait a minute. They didn't need to tell me the mass of the car. That's why semi-trucks can drive next to little VW bugs. It looks like mu is going to be D squared over G. Can I assume that you can crunch the numbers on your own and get the right answer? I don't know what the answer is, but you have the rest of it on there. Is that all right? That way we'll get through more examples. So there's a good example of a horizontal question. Let's do an ask to your one. I know which one I'm looking for. I'll recognize it when I see it. There was a table with something hanging underneath the table. I recall. Oh, I like this one actually too. This is a horizontal circle. Number 18, I believe this is. So what we have here is what we call the conical pendulum. Yes. Is that a horizontal circle? Yep. I know the strength of that. Who cares? The circle is horizontal. What are the forces acting on this mask at the obvious ones? Pardon me? I never put circular force on my diagram. If you can't tell me the name of that force, don't be putting it on there. Tension, absolutely. Like the tension in this room when I just glared at you because I didn't mean to be that hard at you by the way. Those are the only two forces. Watch what happens when I add these together vectorially. How do I add them together vectorially? Add them. Mg plus tension. I know I have to stop right there, Pat. I know I can't go further. I know I can't go shallower because what did you tell me my net force has to be Pat? This is what you were trying to spot. It doesn't appear on the free body diagram. You were saying, well, I know my overall force has to be pointing in which direction. It shows up not on my free body diagram, but on my vector triangle, in this case, for the conical pendulum. Ooh, and it's even going to be Selkatoa. Oh, and they don't even want me to find the acceleration or velocity or period. They just want me to find the force, although if I found FC, could I find V or period? Yes, and we'll talk about that in just a second. Let's see. Do I know the mass? OK, great. Then I can go like this. Oh, that's 39 degrees. So tension next to a vertical line is 39 degrees. Tension next to a vertical line is 39 degrees. See the z? I think that's 39 degrees down there, too. Which trig function relates these to? Tangent of 39 equals opposite over adjacent. FC equals mg tan 39. I'm going to add a part B to this question, because I think another nice extension, given all this information, might be to say part B, find the speed. What's another way to write FC that might have a speed in it? That's an acceleration. Don't forget what you have to put in front of it, please. What if put in front of it? You're giving me an acceleration. No, no, no, even simpler than that. No, no, even. Forces what times what? You're not giving me the mass. You're telling the only acceleration. You're doing this. And that's totally wrong, because you know what's going to happen to those masses, Jafar. You're going to cancel. Don't forget that's not a force on your formula sheet. It's an acceleration on your formula sheet. And if you want to make it a force, you better put a mass in front of it. By the way, am I harping on a common error that drives me crazy? Yes. Let's see. Did they give you the radius? So could you find the velocity? Would that be a fair question? I think so. Or as a part c, instead of finding the velocity, I could ask you to find the period. Same diagram, same information. I would go m for pi squared r over t squared equals mg1039. By the way, up here, let's finish this. I think v would be the square root of rg1039. And I'd go type that in, so you can do that, right? Here, once again, the masses would cancel. Here, I think the period is going to be, let's see. It would move up here. Those would move down. It's going to be 4 pi squared r all over g1039 square root. Yes, I hope I think. That's all hidden in there, too, in the conical pendulum. Let's go back here. What haven't I used yet, that 1.9? Why'd they give it to me? Throw y'all. Well, no, what they wanted to do is they wanted to give you at least one different thing in meters to tempt you to make a sloppy mistake. Instead of the radius, pick the length of the string. So all of that can be contained in that one little question there. Am I right? 4 pi squared r over g1039 square root, right? Yeah. OK. Oh, if you knew the velocity, could you figure out the mass? Maybe. Certainly, if you knew the tension. So another twist, instead of giving you mg, if I didn't give you the mass, but I told you the tension in the cable, you could use this. Except here, I think you'd be using sine or coast or whatever. So don't just lock yourself into tangent. But this conical pendulum, I can do a lot with that as a horizontal circle. Last one, I know what I'm looking for. I know what I want to see it. Show me the table. Maybe 21. I thought there was a yuckier one. Serious, Mr. Duke? You had one on here. Maybe it was in the homework. Let's go do 21 there. I think I said it was 21, right? Let's go find 21. Find the next one. Find this one. There we go. That's what I was trying to get to. Stop there. Stop there. Don't do that. OK, fine. Here's a good example. Horizontal circle again. Oh, in this case, they want me to find the maximum mass. So they're saying this rope can withstand the maximum tension of 6.3 newtons. Going to make it, Mr? You sure? OK. If it goes faster and faster and faster, the tension would go bigger and bigger and bigger. But if it spins at a certain speed with a period of 2.1, what's the biggest mass we can put on there before the stringles now? OK. What are the forces acting on this guy? Get the obvious ones. Gravity down, normal up. How is this moving in a circle? Where must my net force be towards the middle? So there has to be a force here. What? Tension. Tension equals fc. Tension equals jafar. Don't forget the m. And then do I want to use the equation that has velocity or period in it? What do they want me to find in this question? The mass. I got a bit of a problem here. What letter did I use for period? What letter did I use for tension? So you know what? I'm going to do this. I'm going to call it force in the rope. Otherwise, I'm just going to get confused, aren't I? I started to pull a t cubed out of there and go, wait a minute. That's not going to happen. Anyways, the mass is going to be the force on the rope times the period squared divided by 4i squared r. Do I know the radius? Oh yeah, they gave it to me in the diagram. Do I know the period? 2.1 seconds. Do I know the force? 6.3 newtons? Yeah, I can find the mass. Yeah. That m is in front on the top. That moves up. Those move to the bottom. That moves to the top. Well, again, what is on your formula sheet? Acceleration. So if you want a force, force is what times what? You better have that. So yeah, that doesn't appear. Look at a couple of vertical circles. Vertical, vertical. There's two main types of vertical circles. The first is the airplane doing a loop. Because in an airplane doing the loop, your head points down. So here would be an example. A pilot of mass, 65 kilograms, is in an airplane traveling at how fast do planes go? Well, in miles per hour, let's say, let's make it a bit of a jet. Let's go about 400 miles per hour. So let's say about 220 meters per second. The pilot enters a loop of radius 1,100 meters. Find the normal force at the bottom of the loop. I'm not going to try drawing a picture if that's OK. I'm hoping you guys can visualize it. At the bottom of the loop, here's our pilot. What are the forces acting on him at the bottom? Oh, sorry, you guys are still writing. Yeah, I'll get the thing, if people are still writing this. You know what, Tandon? I'll just clean it. Done? Good. What are the forces acting at the bottom? Jimmy, I agree. MG, what else? Why did I draw it bigger? Because at the bottom, the center of the circle is upwards, and he's moving in a circle. Where must the net force be pointing towards the middle? Which force is pointing towards the middle? Normal force. It's got to be the winner, destiny, destiny. So my equation is going to look like this. Winner minus loser equals FC. Again, Jafar, did FC appear anywhere on the diagram? No. It appears on the right-hand side. It's my net force. It's my winner minus loser equals. The normal force is going to be plus the MG over, plus. FC is going to be, don't forget the M. Am I going to use V squared over R or 4 pi squared R over T squared? I've heard two answer. Which one? Y. Yeah, didn't give you the period. Hey, crunch the numbers, see what you get. 65 times 9.8 plus 65 times 220 squared divided by 1,100. I may have made up terrible numbers. It's possible this pilot might die. That's why I want to know. And you guys are laughing at that? You guys have seen way too many movies. You're a violent generation. Do you always get 3,500? I get a normal force of 3,000. Well, I guess if I go to three-sig figs, it's 3.5 times 10 to the third newton. I just want to divide that by 9.8. How many G's is he getting? Yeah. 350 G's. T's more than that. G would be 9.5 to the power of 5.5. Yep. Only 220 squared. Well, I'll apologize and I'll simply say dead pilot. But let's do B. Let's find the normal force at the top. Now at the top, you need to realize if you're in a plane, there's your legs, there's your body. Your head is pointing downwards. So what are the forces acting on this pilot? Get the obvious one. Mg down. Which way is the normal force acting? Down. Two winners, no loser. Mg plus normal force equals Mv squared over R. Equals FC, but I've plugged in the equation already. At the top, the normal force is going to be Mv squared over R. Take away Mg. Conveniently, I have Mg plus Mv squared over R on my calculator. So I'm just going to make the Mg the negative in front. Mr. Goulet, could you please call extension to 00 please? Immediately? OK, I get. Still dead, but you do feel a little bit lighter at the top of the loop if you're on the inside. Now, I called this the airplane. On a roller coaster, are you on the inside of a loop or the outside of a loop? At the top, is your head pointing down or is your head pointing up? Top of a loop, your head's pointing down on a roller coaster. No, no, no, no. Roller coaster doing a loop here. Top of a loop, your head's pointing down. That's a lot. What about a ferrous wheel? OK, so this gives us the second type of question. Ferrous wheel type. I think I have one of those in here. So let's, ah, look at that. Number 22 was a vertical loop that probably wouldn't have killed him. Oh well. I know I have a ferrous wheel picture in here somewhere. 41. I know 16 is as well, but I kept the moon forward. Can I get this on a little page? Pretty close. 75 kilogram person rides the ferrous wheel, which is rotating. And it tells me the centripetal force in this case. It says 45 newtons. OK, a little twist. Instead of finding FC, they're telling me FC, and they want me to find what force the seat exerts on the top or the bottom. Now instead of telling me FC, remember they could have given me the velocity and the radius, and I could figure out FC. Or they could have given me the radius and the period, and I could have figured out FC. In this case, they gave it to me. That's fine. At the top, what are the forces acting on this person? Get the obvious one. Gravity, which way down? Which way is the normal force here? Why do I draw it smaller? Because I know the net force has to be towards the middle because it's moving in a circle. At the top, I have mg minus fn equals 45. I have the normal force equals mg minus 45. I'm going to show you, by the way, we don't even actually need to crunch the numbers for this one. We can figure it out by being clever. Bottom, mg, why don't I draw the normal force bigger because it's moving in a circle and that force has to be towards the middle? Normal force minus mg equals 45. At the bottom, the normal force equals 45 plus mg. Don't get a calculator. Bottom's going to be more because you're adding. Top's got to be less. It's got to be that one. By the way, I would have said that one's wrong. It's not the same in both. I've been on Ferris wheels. If you're lighter at the top, dude, those are vertical circles. Yeah. But I'm a nerd. I think differently for most of you guys. Then we brought in gravity point. I think you guys are OK on fg equals big g, big m, little m over r squared. Be prepared, though, for me to do something like this. Instead of saying calculate the force, what if I give you the force and say, find the mass of the planet? You guys handle that, OK? In other words, so we moved on to gravity. If I tell you that and the radius and the satellite, can you tell me the mass of the planet? Can you solve for that? I hope so. Or if I tell you the mass of the planet, how big the satellite is and what force it's experiencing, could you tell me how far it was from the center of the planet? Then we also added gravitational field strength, which I did a big song and dance about today. I said that was that there, OK? So if they mentioned gravitational field strength and you only see the mass of the planet, don't freak out. Or if I tell you the gravitational field strength and how high we are, could you then figure out the mass of the planet? Or if I tell you the gravitational field strength and I tell you what planet we're on, by the way, don't expect that you're on Earth on the test because an easy way to change a question is to put you on the moon or around the sun or around the planet, ubernerd or whatever I want to and make it my own stuff. Could you tell me how high you are? OK. But what you guys are really wondering about is something like this. So I want this combined gravity circular questions. Let's make a new planet, mass. Let's suppose we have a planet A, J, B, P. No, wait a minute. J, if your last name for some reason. JJPBX has a mass of 6.36 times 10 to the 30, a little bit bigger than Earth, has a radius of 7.2 times 10 to the 6 kilograms in meters. So there's our mystery planet. What can I ask you? Number one, satellite is in orbit. It goes around the planet once every 16 hours. It is its orbital radius, speed, its orbital velocity. When they start asking you, when they start talking about something in orbit and they're talking about orbital radius or period or speed, what we're really saying is that you're moving in a circle and the force that's pulling you in a circle is gravity. What we're really saying is this. Gravity is always big G, big M, little M over R squared. FC depends. Don't forget to write the M. If anybody knows where the Mexican students are at this moment, could you please contact 203 regarding Mexican students that are here in the building, please, 203? Did they give me the speed in this question? The period, except the period in what? Can you tell me how many seconds that is, please? Yeah, you have to go time 60, time 60. Can you do it? Sorry? 16 times 60 times 60. So I'm going to use the 4 pi squared R over T squared. Now, the nice thing is, what do you notice about the little masses? Because I didn't tell you how heavy the satellite was. What are they asking me to find in part A? It's got the R by itself. Now, kids want to cancel an R. No, no. One's on top, one's on the bottom. In fact, when I move both of those guys to there, I'm going to have an R cubed. A T squared up here, a 4 pi squared down there. Right? Stuff moves diagonally. So we'll have this. R cubed equals big G, big M, T squared all over 4 pi squared. I want to get rid of a cubed. OK. R is going to be the cube root of, and this is about as long an expression as you'll end up typing in this year, 6.67 times 10 to the negative 11. The mass of the planet just 6.36 times 10 to the 30th. The period, 57,600? Yeah. Squared all divided by 4 pi squared. I would do what's inside first and then cube root. Can I get a radius of 3.3 times 10 to the 1, 2, 3, 4, 5, 6, 7, 8, 9? By the way, what if instead of asking you for the radius, what if I'd ask you for the height or the altitude? Take that number there. And what? Look too far, Mr. Duke. Subtract the radius of the planet and whatever's left is your altitude. Now, in this case, the radius of the planet is so small and this number is so big, you won't notice much of a difference. But I'd pick better numbers on a test because I just made these up. But for what it's worth, if you wanted to find the altitude above the surface, just subtract your radius from the radius of the planet. Whatever's left is how high above you. B, orbital speed. OK. We would go, well, sorry, Mr. Duke, don't take shortcuts. We would again go gravity equals circular. And I would go big G, big M, little m, all over r squared equals mv squared over r. Once again, that's cancels. This time, one of the r's cancels because there's two on the bottom on one side and one on the bottom on the other side. Let's see, do I know G, 6.67 times 10 to negative 11? Do I know the mass of this mystery planet? Yeah, 6.36 times 10 to the 30th. Do I know r? Hey, what did I figure out in part a? How would I get the v squared by itself? Square root. So we won't solve this. I'm just going to say for what it's worth, it looks like v is the square root of big G, big M, over r. Can you see, though, I could also have given you how fast the satellite was traveling and said, find the radius? Or as a two-step question, given you how fast the satellite was traveling, and said, find the period, you'd find the radius first, and then you'd go to this one, but instead of solving for r, you'd solve for t. But I gave you the period and the questions. In other words, you know one, you can find the other two, whichever one they throw at you. How do I know to use that approach? Have they mentioned in the question, work anywhere? Did they mention energy anywhere? Do you see a change in height anywhere? That's how I know it's forces. And the flip side of that is, by the way, if they mentioned work or energy or there's a change in height, you're probably using energy. Gravitational, potential, energy, energy, energy. Probably this is what you guys are wrestling with. I agree, it's not a tough topic, it's a confusing topic. You really gotta keep stuff straight. Can you remind me at the end of class today, who's in my Block C class? My Block Bs, I went through the review and said, I like this question, I like this question. Did I do it with the Block Cs or did I forget? Can you remind me to do that at the very end of this and we'll tack that on to the video also? I'll give you a few hints since you're here. I defined, that is on your formula sheet. Why negative? Because we said that if you have potential energy, you want to fall. If you're out at infinity, do you want to fall? So how much potential energy do you have? Desmond, can you contact Local 202 please, Desmond? And if you're closer than infinity, you must have less energy than zero. Oh, you have negative. So what kind of questions can we ask you? Well, let's see if I can find a couple. First, I'm gonna expect, I can simply say, find the potential energy at a specific altitude or height or radius. That just crunches the numbers. For whatever planet I give you or whatever satellite we're dealing with. But usually what we're doing with this is stuff that involves work. So let's see, I think now we're gonna try and pull some of these out. Please tell me this was my potential energy loss. Hey, it was. Let's grab some from the home of Mr. Derrick. So let's go, says calculate the orbital potential energy of a 12,500 kilogram rocket on the surface of the Earth. Why do they say orbital potential energy? They're saying don't use MGH please. Use the Cosmic. I would say potential energy equals negative big G, big M, little M over R. Have they mentioned a change in height here or anything like that? No, it's not work. I'm not moving something. G negative nine, sorry. G 6.67 times 10 to negative 11, big M mass of the Earth, little M mass of the rocket, R radius of the Earth, that distance. I'll assume you can crunch those numbers and you can check your answer. I'm more interested in the next couple of questions after that, but I figured it was better if I started with those two. So bear with me, I'm gonna go clip. A 12,500 rocket ship on Earth at rest blasts off and reaches a height of 2,000 kilograms, kilometers above the Earth. Now I got a problem because it's kilometers. I would right away say offer Pete's eggs. Two times 10 to the sixth meters. Ooh, above the Earth's surface. So when I wanna find the radius, I'll have to add the radius of the Earth as well. A says, find the final potential energy. Oh, it just wants final, it didn't say change in. Okay, potential energy final is gonna be negative big G, big M, little M over R final. It's gonna be negative 6.67 times 10 to the negative 11. Are we Earth yet? 5.98 times 10 to the 24th, 12,500 divided by, what's my final radius? Well, it's an altitude of two times 10 to the sixth, but I also have to include the radius of the Earth. That's my final distance from the middle of the Earth. Is that okay, Brandon? Could you type that in and get the right answer? Then I won't worry about it because the answers are there and part of your homework anyways. B, find the work done by the rocket engines. Okay, work. I can't use work equals force times distance because the force is not constant. The further you go from the planet that we could grab it again. So I have to use another definition of work. We said that work was, this is not on your formula sheet. Gotta have it here. We called this the work energy theorem. I think though I can in this case do that because if I'm reading this correctly, it starts at rest, right? So initial kinetic is zero and they're not putting an orbit. They're lifting it up to an altitude and then little invisible angels or whatever are holding it there because if they let this rocket chip go it's gonna fall back down. So I think it's final kinetic. It's at rest, it's also zero. What's changed in anything? So you're gonna get this. Negative big G, big M, little M all over our final, which is two times 10 to the six plus 6.38 times 10 to the six minus negative big G, big M, little M all over our initial, which would actually be the radius of the earth because that's where you started out. What do you see here, Pat? A minus minus, which is a plus. In fact, you're gonna get a positive answer because all the top numbers are the same but this is a way bigger number than that. The negative number will be way smaller because you're dividing by a bigger number. You'll always get a positive work. You should because the spacecraft doesn't want to go up there. You have to supply energy. You have to put in a positive amount of energy to get it up there, the rocket fuel. So you will get a positive answer. Is that okay? So what if you let go of that rocket? Well, in this case, we have a similar question. We have a 1200 kilogram mass and it's released from a height of 4,500 kilometers above the earth. Can I just clue in? I think this is 4.5 times 10 to the six meters. Oh, above, so I'll have to add the radius of the earth. It falls to the earth as no external force acts on the object energy is conserved. It says, first of all, write a work energy equation. I'm not gonna do that. Find its initial potential energy, okay. Negative 6.67 times 10 to the negative 11. I'm gonna try plugging in the numbers right away. Big M, earth, little M, satellite, all over. What's my initial radius? Here we're not starting on the ground. We're starting up in space. So my initial radius is gonna be 4.5 times 10 to the sixth. That's the altitude plus the radius of the earth. That's the total distance from the center of the planet. That there is your initial potential energy and it's gonna be negative. What would its final potential energy be when it reaches the earth? It would be this, but in the denominator, all I'd have was 6.38 times 10 to the sixth. I'm more interested, part D is kind of nice. How fast? Do we have a change in height? Are they also wanting speed? Conservation of energy. Potential energy initial plus kinetic energy initial equals potential energy final plus kinetic energy final. And I'm pretty sure it's starting out at rest because it's released from a height. But I don't think any of the other ones are zero. I think you're gonna have negative big G, big M, little M all over 4.5 times 10 to the sixth plus 6.38 times 10 to the sixth. That's my R initial. The altitude plus the radius equals negative big G, big M, little M. I'm just showing you the numbers that are gonna change in the interest of time. Big G, six, six, seven, eight, big M, earth, little M, satellite. I've scrolled down, but however much it weighed. Your final radius would be the radius of the earth plus kinetic energy, far. So what's kinetic energy final? One half M v final squared, which I'm pretty sure is what they want me to find, isn't it? That's, if that's our final height, that's our velocity at that height. Yeah, that's when it hits the earth. No, no, no, no, no, no, no, no, because the force is changing, gravity is changing. You can't use MGH because we're going cosmic here. No, no, no, no, no, no, no, no, no, no, no, no, no, no. No, no, no, no, no, no. By the way, if you used MGH, you'd get a zero here, which would totally skew the question. You have a question or was that just, okay. Third type of work energy question. So the first one, here's variations on lifting it in orbit. The third type is lifting it and putting it in orbit, which means you're going to give it potential and you're going to give it kinetic energy. See if I can find one. I mean, I've glossed over some details, but eventually you are gonna have to look at your notes and do some homework and I'm going to help you part of this, I'm also bailing out, not, I don't know if it's you guys, but I'm bailing out a bunch of kids that did no homework for the week before spring break and then no homework during the basketball tournament. I recognize that. Not on my helping you guys out a little bit, but you know what? Some people have made decisions that might affect the grade. For me, you made a positive decision, but I'll give you a shovel, I'll give you instructions on how to dig out of the hole, but I won't throw any dirt for you. Okay, so here's a classic. Number five from lesson seven. We'll just talk about it. Okay, by the way, the author here uses RTZAI, that's relative to zero at infinity. What he's saying is use the cosmic potential, not use MDH, right, that's our hint as well. So could you find the potential energy of the satellite on the surface of the earth? Yeah, negative big G, big M, little M over our earth. Could you find the potential energy of the satellite when it is in orbit? Yeah, negative big G, big M, little M over our final. Could you find the orbit speed and the kinetic energy of the satellite? Didn't we a few examples ago when I did FG equals FC? Couldn't I use that to find your orbital speed if I knew the radius? Oh, and once I know the orbital speed velocity, kinetic energy is a half MV squared, yes? Could you find then the total orbit energy? Sure, it would be whatever the potential was up there, plus whatever the kinetic was up there, and it would be negative, because to get out to infinity, you'd have to add energy still, and then find the total work. So let's try this puppy, and I think this might be what we'll finish with, so I hope I think that one. This is a better question, let's do this one. Now I haven't covered all the little curve balls and tricks and whatever. Remind me though to go through the sheet and tell you which questions I like, okay? By the way, so look at this question here, where they give you the mass of the satellite. Once again, he does 5,000 kilometers above. Once again, I'm gonna go, that's five times 10 to the sixth meters. I could ask you E without walking you through everything else. If they just said, hey, how much work? The work to put something in orbit is the change in potential plus the change in kinetic. If you're lifting something and holding it there, your change in kinetic would be zero, and that's why we just did change in potential. But if we're going in orbit, that's a fairly long question. We would need to do change in potential and change in kinetic. Let's do A, first of all. The potential energy on the Earth, I'll call that initial. That's gonna be 6.67 times 10 to the negative 11. 5.98 times 10 to the 24th, 10,000, divided by 6.38 times 10 to the sixth, and except it's negative, I forgot the negative. Woo, that was sloppy. Would you be able to crunch that and get an answer? Then I'll trust that you can. B. What would the potential energy be when we're in orbit? Gonna be the same, the same, the same, all over 6.38 times 10 to the sixth plus an altitude of five times 10 to the sixth. That's the only difference, which means you could probably go second function, enter on your calculator, and just change a couple of numbers and I'll have to retype the whole thing. C says, find the orbit speed and the kinetic energy. Well, first of all, I'm gonna say the kinetic energy is gonna be a half m v squared. They gave me the mass, I've scrolled down, but I'm pretty sure they gave me the mass. Yeah, 10,000. Really, what C is saying is find the velocity, and I would have no problem as a question giving you the same information and then simply saying, tell me the kinetic energy in orbit. And I'd assume that you'd clue in, I gotta find velocity, because that's really what this question is asking. How do we find speed? Do I wanna use the one with speed or period in it? Yeah, v squared equals big G, big M, or let's see, type, type, type, type, type. Oh, and this is actually kinda nice because it looks like the kinetic energy technically is one half M, big G, big M over R, because if I square the v, I just lose the square root. In fact, big G, big M, little M over R, if there was a negative there, that would be half the potential energy, but negative. We said you don't memorize this, but for what it's worth, if you wanna know how these are all interrelated, as it turns out, the kinetic energy of a satellite is the same as its potential energy divided by two, but positive. Just is, because they're so related together. Oh, and then D, total orbit energy. So if they ask for total energy, how does the word total mean to you mathematically? What mathematical operation do you think we do? It's gonna be the potential energy in orbit plus the kinetic energy in orbit. It would be the answer that you got for B plus the answer that you got for C. And there is a shortcut you actually find as it turns out, the total energy in orbit is half the potential energy, but negative. You don't have to memorize that, but if you're doing it and you say, hey, wait a minute, I know that's exactly half of that answer up there, you can probably, I probably did it right, okay? I hardly ever use that shortcut though. I usually prefer on a test to take my time and get it in a nice, comfortable way. And then E, the work, work is change in potential. You go that minus that, plus change in kinetic. Your final kinetic is this bad boy here, your initial, zero, because you're starting from rest. And you add them up and that's your work. And you would get a positive answer because absolutely it requires you to do work to get a satellite in orbit. That doesn't cover all the bells and whistles, that covers the bulk of it. Then you can look at the review and look at my solution key and figure stuff out. Now, questions that I like. From the review, I'm gonna hit pause for two seconds. So a couple of hints, I've already said, please be prepared in the gravity formula to solve for mass or for radius if you know the force. There's a couple on here that I do that. A classic example would be, for example, to say there's a satellite orbiting the Earth. If I know the force that the satellite feels, can you figure out how heavy this, but what the mass of the satellite is? You know the mass of the Earth, I tell you the radius, et cetera. Bear with me, bear with me. I'm trying to find a specific one that I can't. It's a multiple choice, so I'm not gonna freak out if it's a bit tricky, but okay. Let's take with the program, Mr. Dewick. Which ones do I like? You should definitely understand stuff like number one, the conceptual stuff. Which way is acceleration, which way is velocity, which way is force, okay? I think there's a bunch of those in the review. So I'm not just gonna say I like them more, I like, there's gonna be a conceptual question where I ask you about direction. Sorry while I'm hesitating and having to remember. Something like seven is fine, eight is good. I like eight, 13 is nice. Again, 14, another conceptual, like make sure you understand that force and acceleration are inwards, velocity is not inwards, velocity is 90 degrees. Make sure you know those. There's some kind of an amusement park ride that's prepared to handle like 16, a Ferris wheel or something like that, okay? I believe I put a horizontal circle somewhere on here like the conical pendulum, like number 18. And remember I said as twists, instead of giving you the mass, I would have no problem giving you the tension. Or other stuff I can muck around with. Oh, you don't wanna click there. Make sure you can figure out gravitational field strength, remember gravitational field strength was big G, big M over R, so gravitational field strength of Mars or of a planet or of the moon, something like that. 23 is good. I like, make sure you understand how the graph works. Oh, there's what I was looking at. Sure, 24, where I said I'd give you one where I'm gonna either ask you to find a mass or a radius. So here's one where you know the force and you gotta find something else. Okay, I knew I had one on there somewhere. Or even 25, I'm noticing 25, they're asking you to find the mass of the planet as a twist. Instead of saying find the energy, I find the planet. Okay, be prepared for something like that. Oh, that was a lot. Sure, there's one similar, that's scaring me. 42 is good. There's some kind of a question about graphs. Well, there's some hints, anyways. There's others. The test, I'll even give you more specifics. Test is mostly multiple choice. It's 11 multiple choice. And then one, two, three written. There is a using principles of physics kind of a question on it. But I think you'll find the multiple choice probably is the more challenging. But if you've done the review, I can't say this time that you'll have seen every question. You'll have seen every type of question, but be prepared for me to give you one more. Oh, that's backwards from the way they asked on the review. They went forwards, I went backwards. That kind of a thing. Is that okay? Do you have any specifics that you wanted me to do? Or is it better, you know what? Probably better for you to work through some of this on your own now. And then if you have specifics, I'll probably do 10 minutes at the beginning of each class, okay? I'm gonna hit save, first of all.