 Lesson five, orbits. Now we're going to get into some nerdly cool stuff. This is the physics that they were working on in the 1940s and the 1950s when they said hey let's try and think about how can we send up a satellite and then in the 1960s when they were saying how can we send someone to the moon but the moon was not the moon launches were not orbits but certainly in our current society our satellites fairly important to our standard of living to the technology that we use absolutely you guys are used to being able to see or communicate with anywhere on the planet that's only been for about the last 40 years or so up until then to try and reach someone on the other side of the earth we needed to either yet have a physical connection they used to lay long telephone cables across the Atlantic and across the Pacific but those telephone cables could only handle so many calls at once and if they were all being used you could not reach someone in Europe or in another country well with the advent of satellites we just bounce the signals off the satellites to the other side of the planet and it works just fine and it's at the speed of light as an extra bonus example one says draw a force diagram for the mass m little m in orbit what are the forces acting on that little tiny satellite or planet and I'm gonna say get the obvious ones now the obvious one is gravity but which way would gravity be acting towards the big planet the only force acting on it is that one except little note fg is not mg because we're not on the earth surface instead fg is gonna be big g big m little m over r squared but you may notice me again that this planet is also tracing out a path what path is this planet tracing out or the satellite tracing out oprah tell a circle so where is my net force towards the middle and I only have one force towards the middle this is the equation that will never be given to you that you need to realize is occurring when we're in orbit centripetal force equals you know what actually mr. do it right in the correct order gravitational force equals centripetal force because we know this is what's pulling it in orbit and we're moving in a circle we can conclude that fact really it's the same physics as my little ball of string tension toy here here's an orbit except in this orbit string is providing tension is providing the orbital force in a planet or a satellite gravity is providing the orbital force example a 10,000 kilogram satellite is orbiting 20,000 kilometers above the earth you know what this is going to be 20,000 and then three more zeroes meters let's fix that right away right pat we're gonna do one more thing we're gonna underline the word above underline the word above says write the force equations so what we're gonna say for part a here is gravity equals circular or centripetal now gravity is gonna be big g big m little m all over r squared circular I have two options for acceleration I have the v squared over r and then I have the new one that I showed you last day the 4 pi squared r over t squared they're both on your formula sheet which one do I want to use here well is this question dealing with period or speed I glanced at part b and I said oh they want me to find the orbital speed so I'm gonna write this now here's the most common mistake look up kids go oh v squared over r that's not a force what is that acceleration mass times acceleration please but you'll notice something that happens what do you notice yeah the size of the satellite has nothing to do with how high it has to orbit the space shuttle will orbit at exactly the same height and speed as another satellite that weighs only a few kilograms right next to it this the orbit height and speed depends on other things but it does not depend on the mass although they told me the satellite was 10,000 kilograms I don't need that also how many ours on the bottom here how many ours on the bottom here to how many here one of the ours will cancel it if you don't believe me you can move an r up to there and you would say I got one on top two on the bottom one of my ours will cancel and in fact I end up with this v squared equals big g big m all over r now b says find the orbital speed this gets me v squared how would I get v okay v is going to be the square root of big g big m over r and you know what because this is new we're gonna go old school a little bit I'm gonna list my data what's big g what's big g you have to know this 6.67 times 10 to the negative 11 doesn't mean yet to memorize it it's on your formula sheet but you better know how to find it and you'll probably end up memorizing it big m is the mass of the earth I can't remember what the mass of the earth is you know what can all of you get your formula sheets out you're gonna want them in front of you for the next three weeks or so actually for the next month or so because when we do electrons and protons you need to know their masses and their charges and no one memorizes them so mr. Pilgrim what did you get as your mass of the earth please and the last thing that we want is the radius of the orbit now this is why I underlined the word above above means that this distance here is 2 times 10 to the 1 2 3 4 5 6 7 2 times 10 to the 7th but our equation is not designed to work from the earth surface our equation is designed to work from the center of the earth so to this 2 times 10 to the 7th I am going to have to add the radius of the earth which is Justin you have in front of you as well six point something I get so the radius of the orbit is 2 times 10 to the 7th plus 6.38 times 10 to the 6th okay above the earth surface means I better include from the center of the earth outwards is that okay does that make sense here in the out the the other word that they'll use is altitude if they use the word altitude that's also measured from the earth surface if they use the word orbital radius that's measured from dead center and we don't need to add question time 20,000 kilometers which is 2 times 10 to the 7th meters that's that distance right there okay I didn't feel like writing to 0 0 0 0 0 0 0 that okay okay you need to keep straight when you need to add the radius the earth and when you don't if they talk about the orbital radius don't add the radius of the earth please because you'll get totally the wrong answer so what I end up with is this then my orbital speed up there if I want to stay that high how fast do I have to be traveling in my satellite this fast six point six seven times ten to the negative eleven five point nine eight times ten to the twenty four all divided by two times ten to the seventh plus six point three eight times ten to the sixth I'll type what's inside the square root first and then I'll hit equals and then I'll square root answer I think I get 3888 see if you get the same thing yes no yeah 3888 actually three point eight nine times ten to the third meters per second how um how fast is that satellite traveling here on the earth when it starts out zero then we hit the launch how fast does it have to be traveling tangent to the radius to stay in orbit three point eight nine times ten to the third that's a lot of kinetic energy where's the energy come from from the fuel this is why it's so expensive to put satellites up there they're expensive to build yes but they're also darned expensive to put up that's a staggering amount of fuel oh and we haven't calculated how much potential energy you'd also need to give it to get it 20,000 kilometers up where's that potential energy come from the fuel you may notice that any big rocket that you see has large fuel tanks you see the space shuttle right the big underbelly tank which falls off and those two solid rocket engines which are basically nothing but fuel we need that to get the velocity for orbit and the potential energy for it's got to come from somewhere then I said note orbital speed is independent of the mass of the orbiting object it depends only on orbit radius and central mass M so we have a scalar equation we started out by doing this gravity equals circular we said big G big M little m over r squared equals m v squared over r we said hey mass cancels hey radius cancels one of them and we get this now this is the equation for velocity I do not memorize this equation I memorized that I can do this and I can derive whatever they need me to derive if they're talking about the velocity or the speed I'll use this if they're talking about the period how long to go around once I'll use four pi squared r over t squared but there's still gonna be an M in front whatever they want me to find then I can get the variable by itself because this is although it's ugly Megan you'll notice it is straight cross multiplying it is one big fraction equals one big fraction so it's yeah it's ugly but it's not too bad example three says fill in the proof below to show that when an object is in orbit it's inwards acceleration is the same as the gravity field at that distance this tells us that objects in orbit are not floating they're in free fall I'm gonna say that again the space shuttle is not floating in outer space it's falling down it's in freefall and the astronauts when you see them floating inside the space station or inside the space shuttle they are not weightless they are in freefall only they can't tell because the air that they're in is free falling at exactly the same rate as they are so they don't feel the breeze on their face the way I did when I jumped out of the airplane the way I did when I jumped out of the airplane this confuses people like no just to do it I see them floating up there they can't be in freefall because if they were in freefall they'd come crashing to the earth we're gonna come do this proof in a second but here's the explanation this is what Sir Isaac Newton memorized memorized this is what Sir Isaac Newton proposed he said he imagined standing on a large mountain and throwing a ball a baseball or in his case back then a cricket ball very very hard horizontally so throws it pretty hard and it lands there throws it even harder and it starts to go over the curvature of the earth and land there there is a velocity you could throw it at just the right velocity so that even though it's falling it's falling at the same rate as the earth is curving and that there is an orbit all objects in orbit are actually falling to the earth what we do is we give them just the right sideways velocity that velocity at that particular radius so that they're falling at the same curvature of the earth so even though they're falling down they keep moving sideways they say the same distance away from the earth's center yep yep so Pat said mr. do it does that mean we can orbit at tree top level theoretically yes air resistance would be a huge issue nice thing about outer space little or no some of the satellites orbit in the in small parts of the atmosphere because the atmosphere doesn't just stop it just gets smaller and smaller and smaller and smaller weaker weaker weaker weaker but certainly very little air at no air resistance and outer space here on the earth you'd have two issues so the speed you can actually calculate it it's very simple but the radius of the earth right there because if you add one tree to the radius of the earth it's going to be like the ninth decimal place who cares and you can figure out how fast you need to orbit here on the earth's surface air resistance and obstacles so what you really need to be saying is maybe instead of a tree top of a mountain certainly orbiting at the height of Mount Everest is doable okay that's what the astronauts do in the vomit comet by the way you guys know what the vomit comet is what's the vomit comet let me know Dylan or Evan go ahead okay to practice getting used to weightlessness without going into orbit because it costs big bucks I just told you why all NASA has it has a seven I think it's a 747 and it's nicknamed the vomit comet because when you're weightless one of the first thing that happens is you puke your stomach's not used to it but you get used to it eventually and all this plane does is it goes to an altitude of about I think 60,000 feet something like that and it dives down at negative 9.8 meters per second squared and because you're inside the plane and the air inside the plane is moving with you and there's no windows you can't tell that you're diving down all you know is I don't be a normal force anymore I feel weightless called apparent weightlessness not true weightlessness true weightlessness is if you're at the edge of the universe away from any planets and gravity is not acting on you that's true weightlessness apparent weightlessness is gravity is acting on you but we've we're going accelerating at the same rate of gravity as you can't tell okay so let's do this proof them says fill in this proof a equals v squared over r that's inwards acceleration and then it says substitute v equals for orbital speed actually I'm gonna cheat I'm gonna substitute v squared equals because if I scroll back do I have an expression for v squared right here and in v squared over r it is squared so I'm gonna say that v squared I can replace with a big g big m over r big g big m over r and here's what I get them a equals big g big m over r and then divided by one more r this guy which is really Tyler the same as I can see you guys squinting sorry we make it a little larger a better time oh yes I'm sorry that's right you guys got a wonky picture there that's right that's gonna it looks like I finally moved to office 2010 it looks like office 2010 strange enough is not compatible with word from act 1995 which is what these were originally typed in my summer project all these diagrams I'm gonna have to redo them not looking forward to that dividing by r though same as having an extra r in the bottom and you get this is what we said last class the expression for the gravitational field of big g big m over r squared so actually in orbit they are under the influence of gravity they just can't feel it because they're free falling I felt what the astronauts felt when I jumped out of my airplane have I mentioned that I jumped out of an airplane oh the only difference Sally is I felt the wind on my face there they can't because the air that they're in is bottled up inside and free falling with them okay all in orbit is is throwing something at just the right sideways velocity so that it falls at the same curvature of the earth and yeah you could orbit on the earth surface certainly many proposals have been written if we ever build a moon base with no atmosphere there absolutely we could have things orbiting the moon a hundred or two hundred meters up very doable because it's also got such a small radius and such a small mass the speed not very excessive fact numerous proposals have said if we imagine this is the moon and you have another base right here the easiest way to get stuff back and forth would be to build an electric cannon that would launch stuff halfway around the planet easy to do easy to do with with current technology don't even get future technology you need the moon base which is future technology but the actual transport easy okay so the fact that in orbit an object in orbit is in continuous freefall is interesting it means that things in orbit are always falling towards the earth even if they never get there it explains why astronauts seem to float around in the space shuttle both passengers and the ship are in freefall falling at the same rate and therefore the ship cannot exert a normal force against them at n equals zero and it means that astronauts experience continuously until they get used to it the physical reaction or feeling of falling so the feeling if you've ever jumped off of a high dive or the feeling you get on the elevator they have that permanently and they got to get used to it and some of them never do some of the astronauts in the Apollo moon landings for example were throwing up the entire trip now I believe for money now there's a private company that's taking people on the equivalent of the vomit comet and you can experience weightlessness a couple thousand bucks even if I did throw up it would still be way cool you guys see two years ago NASA brought Stephen Hawking the wheelchair scientist onto the vomit comet because he's done so much for physics and they carefully gotta do it very carefully because he's got medical conditions but they allowed him to experience weightlessness and you can just see him just grinning like a schoolboy because it's everything that he imagined that it was supposed to be like Pat since you asked here it is in a cartoon a character kicks a football the wrong way but the football orbits the earth and it still goes through the uprights but it just goes through the uprights in the opposite direction what speed was the football moving or to your question that how fast would you have to be moving to orbit let's say at treetop height okay I know we already have the expression in the box that expression in the box is not on your formula sheet so I'm going to start out by saying we're moving in an orbit so gravity equals circular gravity is big g big m little m all over r squared equals circular is going to be m I got two circular equations is this question asking or talking about the velocity speed or the period speed oh so let's use the squared over r what if instead it asked how long would it take you to go around the earth oh I would use for pi squared r over t squared and you can calculate that as well how long would it take you to go around the earth if you were in orbit however they wanted speed v squared over r low and behold the mass cancels low and behold one of the r's cancels and I get this v equals big g big m over r square rooted 6.67 times 10 to the negative 11 mass of the earth was 5.98 times 10 to the 24th is that right yeah divided by and now we are on the earth surface so let's use the radius of the earth and it was 6.38 times something 10 to the what 6th and square root that whole thing what is my v orb my orbital velocity I think from what I recall I think it'll end up being in the tens of thousands of meters per second I don't think it's in the hundreds of thousands what'd you get doing seven oh that might be right is someone else seven thousand nine hundred yeah okay seven thousand nine hundred meters per second basically eight kilometers a second fast but actually doable if the earth was a vacuum we could easily do that right now we got stuff that we can get going that fast the space shuttle was faster than that once it gets out in the outer space we can do it it's just air resistance would be a huge issue and that's going to be the answer to part C when it says why can't we do this on the earth but why can't we do the air resistance going that fast your object the friction from the air would get you so red hot I'm sure you start to be almost the melting metal right what does be want oh this does want the period you know what I'm going to start out exactly the same way gravity equals circular gravity is still big g big m little m over r squared circular is still mass times AC except I don't want to use v squared over r for AC I want to use the one that has the period in it Megan what's the one that has the period in it you have the sheet in front of you which is why I picked on you know the formula sheet right there find circular motion yes there's two equations for circular acceleration and gravity right okay we're on the formula sheet find circular acceleration there's two equations I want the one with a period in it yes okay by the way I'm doing that you guys need to know that it's there last year for some reason I had half my kids I didn't know what that was or that it was it's on your sheet you don't have to memorize it what is it Megan for again most common mistake kids forget to put the mass there in which case this mass doesn't cancel and since I didn't give you the mass they freak out no it's mass times acceleration now second most common mistake look up this is wrong that's wrong because where would this r end up moving when I move it to the other side on the bottom in fact I'm gonna end up with an r to the third let's get the t squared by itself I'm gonna move the t squared up here these down here this up here this is where this is why by the way at the beginning of the year I showed you that whole things move diagonally trick it's for this equation we get this t squared equals four pi squared r cubed all over big G big M how will I get rid of a squared square by the way what's the exponent on the R if I got the R by itself how would I get rid of a cubed cube root you will be cube rooting sometimes during this unit as well that's if you know how long you want the satellite to take to go around the earth and you want to know how high do I need to put it up there because every different altitude will take a different length of time to go around the earth so if I heard you correctly t is gonna be the square root of whoo for pi squared 6.38 times 10 to the 6th cubed have I mentioned that you probably want to be practicing with your calculator during this unit are you starting to see why hello hello hello divided by 6.67 times 10 to the negative 11 5.98 times 10 to the 24th that's how long it would take to go around the earth if you were orbiting on the earth's surface for pi squared times 6.38 times 10 to the 6th cubed divided by bracket 6.67 scientific notation button negative 11 times 5.98 scientific notation button 24 close bracket oh square root mr. duke square root that and I'm getting 5,070 seconds and you know what I'm a nerd divide by 60 84 minutes to go around the earth is that fast yep if you didn't get that and you want me to come look at your calculator now is your chance to ask because I see about eight of you leaning over to a neighbor showing them your calculator screen screen and you have tears in your eyes two of you seem to have the same name so if you need some help now is your chance to ask you off by a half a minute see Pat what do we say why can't we do this on the earth surface but why is it possible on the moon air resistance and also man-made obstacles buildings things where again in the moon as long as you were over the tallest mountain on the moon you just fine okay and you can extend that also to other planets by the way oh no no Mars has an atmosphere doesn't it never mind most of the planets have atmospheres so also you'd have air resistance on the other planets as well I'll take back what I just said supposing that you want to build a satellite let's say a spy satellite and you want it to orbit the earth every 300 minutes now first of all what's wrong with 300 minutes okay what is that in seconds let's write that right above it how many seconds is that 1 8 0 0 18 thousand that right okay so you wanted to orbit the earth every by the way what is 300 minutes in hours what's 300 minutes in hours please do some arithmetic poisoning Tyler okay so the US or the Russians or China or whoever has the technology they're building a satellite and they'd like it to they figure if we have it go around the earth every five hours that gives us a pretty good spying window how high must that satellite be because there's only one height well that will work well we're in orbit so that must be true big G big M little M over R squared equals M which circular acceleration equation am I going to use ah I think they've mentioned the period here not the speed so I'm gonna be using the one that has the period in it which was what Megan my angel yeah keep going no it's not for pie okay so common mistake number three I was soon as you said that I good she did it because I want to jump on that one kids forget the squared on the pie because you're just not used to square in a pie usually in all the rest of your equations pie has been all by itself and pie squared good what do they want us to find how high now here's what we're gonna do we're gonna find the orbital radius that's the distance from the center of the earth to the satellite and then once we have that we'll subtract the radius of the earth and that should give us the altitude that's left that makes sense in other words the R by itself well let's move this up to there it's gonna give us an R cubed let's move this oh wait a minute yay masses cancel that's good because it didn't tell me how heavy the satellite weighed in fact I think I'm gonna get this R cubed equals big g big m t squared all over 4 pi squared so Sally like if you want to you can memorize all these different permutations I think I've done five so far we have one for v one for team got one for art I know start with this and I can cross multiply and get whatever I need to by itself it's a much better way to do this oh how do I get rid of a cubed oh cool we don't do that very often so it's always nice when I can say hey this is actually a useful mathematical operation the radius is going to be the cube root of 6.67 times 10 to the negative 11 big m what planet are we orbiting okay by the way this is also then let's suppose you're designing a satellite to go explore Venus and you'd like to orbit Venus every five hours here it is or Jupiter here it is this tells you the height right here it's earth 5.98 times 10 to the 24th we want every 18,000 seconds squared all divided by 4 pi squared Justin how many numbers are in the bottom of my equation here to I'm gonna put this in brackets okay you know what how about all of you that are struggling on your calculators type with me this is how I would type this okay I'll put the top in one big bracket so I'm gonna go open bracket six point try that again six point six seven times 10 to the negative 11 come on as I'm trying to demo this thing that are going wrong there we go then instead of bracket bracket times less typing 5.98 times 10 to the 24 times 18,000 squared have I finished the top close it off divided by open a new one four times pi squared close off the bottom I think that's the best way to type it because it's clear and it's also the least amount of typing now we aren't done that's what's inside the cube root the only risk with this is you forget to take the cube root but hopefully you'd say that seems a little large that's like five way bigger than the earth's radius cube oh first cube root on the TI's ah yeah math option for answer and I get a radius of 1.4 well let's see 1.4848 times 10 to the 1 2 3 4 5 6 7 1.4848 times 10 to the 7th meters that's five state fakes mr. do it because that's not my final answer that's not what they asked they did not ask me for the orbital radius they asked me how what how high so if you imagine there's the earth we have calculated we have calculated that distance that's r we want this distance that's height h what is this distance why that's the radius of the earth so I think if I take this answer our orbit minus our earth that's gonna give me my h and I'll put in brackets in our notes the other word they'll use is altitude that's gonna give you your orbital altitude h equals well I still have this number on my calculator minus what was the radius of the earth 6.8 point 4 7 times 10 to the 1 2 3 4 5 6 of course I imagine if you're actually putting an orbit you're way more accurate on your sig figs because I imagine even that much probably makes a bit of a difference who uses GPS only a couple of you the rest of you have never used GPS before what is GPS what does it stand for you're right okay it is I think 36 satellites and they're all fancy word geostationary they all stay above the same point on the earth how do they do that their orbital period is exactly what which what is the same as the earth 24 hours in other words if you change 24 hours to seconds you can figure out exactly how high every single one of them is they're all the same height have to be a geostation it's a geosynchronous or geostationary satellite means it stays above the same place which is very very handy oh your orbital period 24 and they're all at the same height as it turns out well you can see because if that's the same the mass of the earth isn't gonna change grab gravitational constant the average is gonna change I is not gonna change yeah then they all have the same answer no matter what their mass was they're all at the same time get a little crowded because there are many many times we want geostationary satellites get a little crowded turn the page so for what it's worth we did this fg equals fc we did big g big m little m all over r squared equals m for high squared r over t squared and we had two equations come out of here you could either say t squared equals oh masses canceled t squared equals for pi squared r cubed all over big g big m or if we got the r by itself we had r cubed equals big g big m t squared all over for pi squared do I memorize those I don't even memorize this I know that if I'm in orbit and they're talking about period or radius and forces fc equals fg go to it yeah do the masses cancel every time in forces yes but what we're also going to be doing next day is asking how much energy refuel does it take to get up there and oh I guess it would take more energy to put a heavier object up than a lighter object up and they're not gonna cancel them homework number one hey why don't satellites fall down because we give them just the right sideways speeds they are falling down but they're falling down at exactly the same curvature of the earth beautiful love brilliant idea anyways let's go number three pat I know you were wondering about the football question on the moon so we'll move it to the moon here how hard we have to kick a ball on the moon to get it to go around on the moon surface in orbit number five pat check your units kilometers in number five okay number six they're giving you the period but they're giving it to you in hours please change it to seconds gotta do a geosynchronous one number seven number nine astronaut Sally stands on a bathroom scale when in orbit what does the scale read and convince me number 10 and number 11 okay and the take-home quiz which I really should have handed out a few days from now but that's okay take-home quiz last question is a bonus what have I got for orbit somewhere