For more information, please see full course syllabus of High School Physics

For more information, please see full course syllabus of High School Physics

### Gravity & Orbits

- The force of gravity is based off of the mass of the objects involved (m
_{1}, m_{2}), the distance between the objects (r), and the*universal gravitational constant*(G).| →F

g| = G · m _{1}·m_{2}r^{2}. - The universal gravitational constant is
G = 6.67 ·10 ^{−11}N ·m ^{2}kg^{2}. - The force of gravity acts on each object equally, and the direction of the force is towards the center of the other object.
- A
*gravitational field*is an area we can treat as having a constant acceleration. Like the surface of the Earth, in some places the force will only change a negligible amount in the area around the object's location. - We denote a gravitational field with a
_{g}. Thus, for an object of mass m in the field, F_{g}= m ·a_{g}. - If we want to find the gravitational field for a given object with mass M at distance r, it is
a _{g}= G ·M r^{2}. - If something is in orbit, it must have a centripetal force to keep it in the orbit. Gravity provides this centripetal force. For a simple circular orbit where one object is much more massive than the second object, it follows the relation
G · m _{1}·m_{2}r^{2}= m _{2}| →v| ^{2}r.

### Gravity & Orbits

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Law of Universal Gravitation 1:39
- Law of Universal Gravitation
- Force of Gravity Equation
- Gravitational Field 5:38
- Gravitational Field Overview
- Gravitational Field Equation
- Orbits 9:25
- Orbits
- The 'Falling' Moon 12:58
- The 'Falling' Moon
- Example 1: Force of Gravity 17:05
- Example 2: Gravitational Field on the Surface of Earth 20:35
- Example 3: Orbits 23:15
- Example 4: Neutron Star 28:38

### High School Physics Online Course

### Transcription: Gravity & Orbits

*Hi. Welcome back to Educator.com. Today weâ€™re going to be talking about gravity and orbits. This will also complete our first section, how we have now covered all of the basics of mechanics.*0000

*Weâ€™ll start to move on to other things in future sections, but this is it for mechanics.*0010

*We got a good strong understanding of mechanics, you should be proud of yourself.*0013

*So, basic introduction to gravity. Whatâ€™s holding you to the Earth right now? Whatâ€™s holding me to the Earth right now? Gravity, gravity is holding me down, keeping me on Earth.*0018

*What causes the Earth to orbit the sun? Gravity. The reason why the Earth goes around the sun. Gravity. The reason why Jupiter goes around the sun. Gravity. The reason why everything is moving around everything? Gravity.*0026

*Gravity is one of the basic forces of the universe, really, really important.*0035

*Gravity pulls massive objects together. The more mass you have, the more pull you exert on other massive objects you pull.*0039

*Two objects, if theyâ€™re both really massive will pull together more than if one object has the same mass but the other one has a small mass.*0046

*Also, the distance between them affects it. If youâ€™re really close, youâ€™ll wind up having more gravity than if you are very far away.*0054

*Now, this idea of gravity, weâ€™re used to it. We accept it right now, but keep in mind, the idea of gravity, was actually really once stridently fought against.*0060

*People did not believe in gravity at all, did not accept the fact that the heavens, that the stars above us would wind up undergoing the same pulls that we were used to in our finial normal, normal day life existence just on Earth.*0068

*The fact that humans are pulled on the same thing as on the stars seems ridiculous to some people, but itâ€™s the true.*0082

*Everything ends up having the same set of rules.*0088

*Gravity, itâ€™s actually a real thing and weâ€™re used to that but keep in mind people didnâ€™t always think that.*0091

*Law of universal gravitation, formula for how gravity works.*0101

*To derive a formula for the force of gravity, thatâ€™s kind beyond the scope of this course, so weâ€™re just going to start by plucking it out of thin air.*0105

*The force of gravity, the magnitude of the force of gravity is equal to G, some constant, times the mass of first object, times the mass of the second object, divided by the square of distance between those objects.*0114

*Itâ€™s an inverse square. If youâ€™re farther away, itâ€™s not just the distance divided, itâ€™s the distance squared divided.*0126

*Letâ€™s talk about things more specific. Force of gravity, the size of force of gravity is equal to GxM1xM2/r2.*0134

*M1 and M2, two mass of the objects involved, thatâ€™s pretty simple. R, is the distance between the two objects.*0143

*If you really want to be specific, itâ€™s more accurate to say itâ€™s the center of the mass for each object.*0149

*Now, keep in mind, weâ€™re normally going to be dealing with very large distances and comparatively very small objects.*0155

*Like the distance between the Earth and the Sun is considerable larger than either the size of the Earth or even the size of the Sun.*0162

*We can worry about the center of mass but for the most part weâ€™re going to be dealing with such large distances in any case, we donâ€™t have to worry that much about the distance between them versus the distance between their centers of mass.*0170

*Donâ€™t worry about it too much, but keep in mind there is a slight difference there.*0182

*G is the universal gravitational constant, which is the thing that makes this formula run.*0185

*The idea is mass times mass divided by the square of the distance.*0191

*Thatâ€™s what affects it, but we need to have a specific thing thatâ€™s going to let us generate actual numbers and this is scaling factor that actually lets us get numbers by multiplying these things and dividing.*0196

*6.67x10^-11. Newtonâ€™s times metered squared divided by kilograms squared.*0207

*Because remember we want in the end to get Newtonâ€™s out of this.*0214

*We want to get four sides of this, so if we got masses up top then weâ€™re going to have kilograms squared up top.*0218

*That will cancel out there. If we got on the bottom; meters. Then we got to cancel it out up top.*0226

*We got meters squared times kilograms squared, so it will cancel out the two masses and dividing by a distance squared.*0232

*That leaves us with Newton's. That's why the law gives us a unit that seems so bizarre.*0241

*Now notice that G is a really tiny number. G is just incredibly small.*0249

*The reason that we don't feel a pull from buildings around us is because G is so small.*0257

*We're comparatively way closer to a building than the center of the Earth.*0263

*That building has so much little mass compared to the Earth as a whole and we'll talk about the mass of the Earth later on.*0269

*It's a big number, somewhere on the scale of 10 to the 24th.*0277

*It is big. It's really big. That building, it just doesn't have the mass to compete with how tiny that number G is.*0281

*Unless you are an absolutely giant thing. Unless you're basically a stellar body, you're just not going to have the ability to have effective powers in gravity.*0287

*Finally, force gravity is a vector. You have to remember it points between the two objects.*0297

*Object 1, Object 2. Object 1 gets pulled towards Object 2, just like Object 2 gets pulled towards Object 1.*0303

*Equal and opposite reactions; Newton's third law still applies. So the two objects pulled towards each other.*0312

*Gravity is not just a number, it's a vector. You have to have a direction to go with that size.*0319

*To go with that amount of force. Remember that it's always pulling towards the other object.*0325

*Normally we'll be able to treat this as if it's single dimensional, but if you needed it would be actually vector quantity.*0331

*So previously, we simply thought of gravity as a general acceleration.*0339

*We knew G was equal to 9.8 meters per second per second.*0344

*Now we're talking about universal gravitation. So what does that mean?*0349

*What does that make our old conception of 9.8 per second per second into?*0351

*Such an acceleration, we call a gravitation field.*0356

*We know that this is still valid and useful and worthwhile because we can actually model lots of real things with 9.8 meters per second per second.*0359

*It works, we've probably by now done a few labs or at the very least we've done so many examples that make intuitive sense that we see that 9.8 meters per second per second is actually is pretty reasonable thing.*0367

*The world pretty much runs on that.*0378

*How do we make these two come together?*0383

*A gravitational field is a way of saying at a certain distance, you're going to experience a certain acceleration.*0385

*How can we find gravitational fields in general?*0393

*A gravitational field imposes a constant acceleration on anything inside of it.*0397

*Remember before, we had force of gravity equal to the mass times the acceleration of gravity.*0401

*Force of gravity equals the mass times the acceleration of gravity.*0406

*For now, any object. This will work on Earth, but it will also work on anything.*0410

*We saw this before with the force of gravity on Earth, but we can do this on Mars if we knew what the things involved were there.*0414

*We could do it on the surface of the Sun, we could do with any object that we felt like.*0421

*Connect that formula with the law for universal gravitation. We're going to have that force of gravity is going to equal the mass times the acceleration of gravity on one side and gravity times M1 times M2 over R squared on the other side.*0426

*For example letâ€™s talk about me. I will consider myself to be one of the masses.*0438

*I'm Mass 1. I'm M1. I'm M1 times acceleration of gravity, is the force of gravity currently pulling on me.*0449

*From universal gravitation, we also know that the force of gravity currently pulling me is G times M1, my mass, times M2, the earths mass, divided by the distance between my center of mass and the earths center of mass.*0455

*The distance between here and the center of the Earth.*0467

*Mass times acceleration of gravity equals G times M1M2 over R squared.*0471

*That means my M1 and the M1 of the universe of gravitation cancel out and we're left with the acceleration of gravity is equal to the mass of the object we're looking for the gravitational field of times G divided by R squared.*0476

*However far up we're putting our gravitational field.*0489

*So in case of the Earth, for me standing up here talking to you. The distance I'm going to get, whether it's here or I climb a mountain, or I dive into the sea.*0492

*I'm going to change my distance by a kilometer, two kilometers. The size of the Earth is so much larger than that, that my change in R is a drop in the bucket compared to it.*0505

*While the exact, the precise amount of gravity that is affecting me will change slightly.*0517

*Its going to change a negligible amount. Which means that gravitational fields will work when we've got a very large, very massive object.*0522

*The distance we're going to get from that object center point is very little compared to the distance of the whole thing.*0528

*Our change in distance is going to be so small compared to the full mass of the distance that we can basically treat it as a constant acceleration as opposed to having to re-calculate the force at all times.*0537

*That's why G equals 9.8 meters per second per second worked, because no matter where I'm going to go on the surface of the Earth, I'm really not going to get very far from the surface of the Earth.*0547

*Unless I'm getting in a space ship. We can treat it as if I got a constant acceleration because R just is not going to change that much and everything else is going to remain constant.*0556

*In orbit. Orbit is one body rotating around another.*0567

*From our work in uniform circular motion, we already know to be in a circle, the acceleration has to be equal to the speed squared divided by the radius of the circle and that immediately gives us that the force to cause that to happen.*0570

*The force is equal to the mass times speed squared over the radius.*0583

*So what is the centripetal force that keeps a celestial body rotating? That keeps celestial bodies rotating each other.*0586

*What would that force be? Gravity.*0596

*If the objects have no other forces acting on them, which makes sense if we're in deep space or we're fairly out in space and we don't have to worry about other things pulling.*0601

*We're moving in a circle and then we get force of gravity is equal to force centripetal, which we can expand into the gravity times M1 times M2 over R squared equals M times speed squared divided by R.*0607

*One thing to point out, this isn't just M. It's M1 or M2, depending on which one we want to make it.*0620

*The object that's moving around, the M1's are going to cancel out on either side.*0626

*The other thing to note is that I want to point out that in real life, orbits are almost never circular.*0632

*Orbits can be close to circular but normally orbits are actually elliptical.*0638

*A circle is something that has a constant radius. An ellipse is something that is able to squish out.*0643

*An egg is kind of an ellipse. Things that get squish.*0648

*An ellipse is something that, we can have an object that can go around in an ellipse or it can go around in a circle.*0663

*We've been dealing with circles because they're much more sensible, much easier to work with, but in real life orbits are actually ellipses.*0670

*Also, in real life, when the Earth is going around the Sun, there is something else working on it.*0676

*All these other planets around us. Now comparatively the Sun is Big Pop in our universe.*0682

*The Sun, it's got the most mass by far. It's able to have the most effect on our orbit.*0687

*There is a whole but of other planets out there. One of the important planets that also has a really big mass, Jupiter.*0692

*Jupiter has a really large mass compared to the mass of Earth.*0699

*It's able to also have some effect on our orbit. Very little compared to the effect of the Sun.*0702

*If this real life, if we want to as correct as possible, we're actually dealing with an ellipse, we not technically dealing with a circle.*0709

*We're actually having to deal with other stuff, we're not having to just put this in a vacuum of force of gravity, one force of gravity is equal to the centripetal force.*0715

*There's more stuff happening here. At the same time, we don't have to necessarily worry about it to be able to get pretty good answers.*0722

*Just like when we were like 'technically there is air resistance, technically there is the other things when dealing with objects falling' at the same time, we can normally still forget air resistance and be able to get lots useful answers.*0730

*Except in really egregious cases where it's moving really fast, we have to clearly care about it.*0742

*In this case, it's one of these things were not it's a really egregious case. The mass of Jupiter is comparatively little to the Sun.*0747

*We don't have to worry about the fact that we're not going to calculate with it if we wanted to figure out something between the Sun and the Earth.*0755

*At the same, if wanted to be really rigorous, we would have deal other calculations and make it a whole lot harder.*0761

*So like air resistance, we kind of put in on the table, left it for a later physics course.*0766

*We're going to wind up doing the same thing with weird orbits that are not circular and other forces of gravity operating, but it's important to remember that there are other things out there.*0770

*One really cool idea before we get started in our examples. A famous thought experiment that Newton put forward. Isaac Newton, gives us another way to think about gravity and orbits.*0780

*Imagine, and before I get too far, I would like to apologize for the bad drawing of this Earth, I am terrible at drawing.*0789

*If you live in Morocco or Tangiers or anywhere in the north of Africa or England. I'm sorry, I have basically ruined your place on the Earth. It's just kind of not there.*0797

*This guy is supposed to be Greenland, so if you're in England, my apologizes, if you're in Morocco or Tangiers or any of the other many places that I ruined with my poor artistic ability. I'm sorry.*0808

*Now moving on. Imagine a very tall mountain on Earth. So tall as to be above the atmosphere.*0821

*There will be no air resistance, so we don't have to worry about friction slowing down the object.*0827

*Great. On top of this mountain, we'll put a cannon and we'll fire cannon balls out of it with greater and greater velocities.*0831

*What's going to happen as those velocities increase?*0837

*Let's start doing it, we'll play around with it. Here is the center of the Earth, we shoot something out, stuff is going to get pulled towards the center right?*0840

*That's how gravity works. Let's say we put the cannon in and we practically don't shoot at all, we just let the cannon ball roll out.*0847

*The ball comes out and boom, falls right into the Earth. Well what if we put it would with a slight amount of force.*0854

*Its going to shoot out, then itâ€™s going to fall into the Earth.*0859

*What's going to happen if we put more force? It's going to shoot out...and then boom, it's going to fall out, because it's getting sucked into the center of the circle, remember?*0864

*At every point on the circle, it's getting pulled in. Well if we shoot it harder, it's going to shoot out...*0873

*It'll get pulled in and then it lands eventually. But, if we shoot it really, really hard. Let's say if we shoot it at super extreme, it'll get shot out, it'll get pulled slightly by that and then it'll just fly off into space.*0882

*It'll just go off forever. If it goes off forever, we've lost it, there is nothing there.*0896

*There's not nothing there, it's gone into an escape velocity. It's managed to get pulled far enough away from the Earth that it'll manage to escape the gravity of the Earth.*0903

*If we shoot at the right speed, instead of falling into the Earth or falling out or away from the Earth.*0912

*It's going to get pulled in and it's going to fall into the Earth forever and ever and ever.*0921

*It's just going to keep spinning around the Earth because it's getting pulled in at all moments. So it just keeps going.*0932

*The best one, the one that will be in orbit is a permanent fall. So the permanent fall is way to think of gravity.*0941

*Gravity isn't just pulling a thing directly in, it's a way of thinking of a fall. An orbit isn't something that's not falling, it's something that's falling at just the right rate.*0950

*It's falling in such a way as to constantly miss the ground. Flying isn't necessarily not falling, it's missing the ground as you fall.*0962

*The important thing is that it's still being effected by gravity, but instead of being pulled into the ground, it's getting pulled towards the ground but it's moving fast enough forward that it just keeps going around and around and around.*0970

*This a thought experiment. Thought experiments are a class ideas where you can, instead of having to actually do a physics problem, because clearly you're not going to be able to go up and build a mountain so high up and put a cannon on top of it and shoot it so fast, that's not really plausible.*0983

*We can think about it from all the ideas that we know we can trust at this point. All the things that we've learned so far, we can test that on an idea and come up with all sorts of things.*0999

*That's what a thought experiment does and lots of modern physics and other stuff previously is developed by that, and that's basically how we all do puzzles, we know what we know and we work around it and we're able to come up with all sorts of things.*1007

*That's exactly what this is, it's a thought experiment that lets us understand a cool thing about the way the world works.*1018

*Onto the examples. Two objects have masses of 4.7 times 10 to the 7th kilograms and 2.0 times 10 to the 9th kilograms.*1028

*If the centers of mass are 850 kilometers away from one another, what's the force of gravity between them?*1036

*This is just a really blunt use of the force of gravity formula.*1041

*Universal gravitation, we know that the force of gravity, the magnitude of the force of gravity is equal to G times M1 times M2 over R squared.*1045

*Throw in all the numbers we have. We have 6.67 times 10 to the -11th.*1055

*I'll tell you right now, you just have to memorize that. You're just going to have to write it down and keep it on a card with you or you're just going to have to keep it in your brain.*1060

*Like we need to keep 9.8 meters per second, if you got a lot of gravity problems, you're just going to have to know it. It's something that's just an important number to remember.*1068

*It's one of the basic fundamental concepts of the universe, so it matters.*1076

*6.67 times 10 to the -11th times 4.7 times 10 to the 7th kilograms times 2.0 times 10 to the 9th kilograms.*1081

*So mass object 1, mass object 2 divided by 850 kilometers, wait a second, standard units.*1092

*What's the standard units here? Is kilometers standard? No.*1100

*When we dealt with G, G required M squared. Remember it was M squared over kilometers squared times Newtonâ€™s.*1104

*That M squared, we're going to have to be working in meters.*1111

*Remember if somebody gives you a unit and it's not in SI units, it's not in normal metric units. Change it, change it to a normal metric unit.*1114

*Otherwise things can go so very wrong. Sometimes it will work out, some of the easier problems will work out just fine and you'll be able to keep it that but if you want to be able to really trust what you're doing and be sure that it will work out, change it into SI units.*1122

*Do the problem in normal metric units and then at the very end convert back to the unit that they gave you.*1135

*That's the best way you want to be sure of it. If you get really used to doing lots of things, you'll start to catch more stuff, but really you want to get used to using metric units.*1140

*If you want to be a scientist, if you're going to do a lot of physics, if you're curious about living anywhere else outside of America, you're going to have to do that.*1149

*To all of my viewers outside of America, you're probably not going to have to worry about other units.*1157

*If you live in America, you might want to consider getting more used to metric units and just get a feel for what they're like.*1162

*Anyway. Back to the problem. 6.67 times 10 to the -11th times, etc., divided by 850 kilometers, so what's that in meters.*1168

*So 850 kilometers. 850 times 10 to the 3rd, because it's kilometers; kilo 1,000. So 10 to the 3rd is a 1,000.*1175

*Then we have to remember it is R squared. You pop all that into your calculator and what do we get? Some big giant number?*1186

*No, no, no. We get this here, which is tiny. Then we also got this here which also going to make it really small, we get 0.00009 Newtons.*1193

*That's right, 900,000ths. 900,000ths of a Newton is how much they manage to pull on one another.*1207

*These are fairly massive objects that are at distance that we wouldn't think is that huge.*1220

*Keep in mind that's why this stuff is so, that's why we don't really experience gravity other than the gravity of Earth, because most of the other stuff just doesnâ€™t have that much effect on us.*1224

*Example 2. If the Earth has a radius of 6.378 x 10^6 meters and the mass of 5.974 x 10^24 kilometers.*1236

*What is the gravitational field on the surface of Earth?*1245

*Remember, the gravitational field way for us to go from knowing what the force of gravity was to something telling us about the acceleration of the gravity at a certain distance away from a place.*1248

*Force of gravity is equal to, we want to be something, mass times acceleration of gravity.*1258

*We know force of gravity, we can change this into our general one. G M1 M2 over R squared equals mass.*1265

*Lets make it mass 1 times acceleration of gravity.*1273

*If we have an object of mass 1 on the surface of the Earth, then it's going to wind up having a force of gravity that's G times M1 M2 over R squared.*1277

*But we also want it to have this acceleration of gravity, something else that allows us to come up with a gravitational field for it.*1286

*If that's the case, we got M1's cancel and we get G M2 over R squared equals the acceleration of gravity.*1293

*We plug in all those numbers we know, we get 6.67 x 10^-11 times what's the mass of the Earth, 5.974 x 10^24 kilograms divided by 6.378 x 10^6 meters, remember we're in meters.*1302

*We've got to remember we need to square it when we replace it. So we punch all that into a calculator and what do we get out of it?*1334

*Eventually it simplifies to 9.795 meters per second per second. Hey, that makes a lot of sense.*1340

*What do we normal use? 9.8 meters per second per second.*1352

*So this turns out to be something that works out well.*1356

*Now there's a couple of simplifications that we made doing this problem.*1359

*The Earth does not actually have the radius of this, because the Earth is not actually a circle.*1362

*The Earth is slightly oblong, it's not quite a perfect circle.*1366

*So when you're dealing with it, we don't actually don't get the chance to deal with it as a perfect circle, so we made this problem a little bit easier on ourselves.*1371

*Also we don't necessarily have that the center of mass for the Earth is precisely in the center of the Earth.*1378

*That might be the case, but we haven't been guarantee yet, we need to find out more about the composition of the Earth.*1386

*So there's more things to keep in mind here.*1391

*On to the next problem. So assume that Earth's orbit is circular. Once again it's one of those things we said assume we can disregard air resistance.*1394

*If the Sun has the mass of 1.9898 x 10^30 then that's also a big number.*1403

*1.496 x 10^11 meters from the Earth, what velocity does Earth orbit the Sun at?*1409

*How long does it take for the Earth to complete one orbit?*1415

*We know that the force of gravity, because it's moving in a circle. Is there any other forces operating on it?*1419

*No, we know that it's just centripetal force pulling. Just gravity pulling, so that must be the entirety of our centripetal force.*1424

*We have to force of gravity equal to the centripetal force. Once again, there are other things in the solar system but Sun is big poppa.*1432

*G times M1 M2 over R squared equals, what's the centripetal force, M1 V squared over R.*1440

*M1 in this case is the object moving around. It can be canceled.*1450

*We've got that G M2 divided by, let's multiply both sides by R, equals V squared.*1455

*Now we'll take the square root and we'll get G M2 over R equals V.*1464

*So we can toss all these things in. We get the square root of 6.67 x 10^-11 times the mass of the Sun, 1.989 x 10^30.*1469

*All divided by the distance between the Earth and the Sun square, sorry not squared, because we managed to cancel out those R's.*1487

*1.496 x 10^11th.*1495

*Now we take the square root of that whole thing and after a whole bunch of calculating, we get 29,779.3 meters per second.*1504

*The Earth is really whizzing through the solar system.*1516

*Now if we want to find out how long it takes for the Earth to complete one orbit, how far a path does it have to travel?*1519

*Well the circumference, the path it has to follow is 2Ï€R right?*1525

*So the orbit time is going to be, how far a distance it has to go, 2Ï€R divided by the speed that it's moving at, V.*1532

*We start substituting in the numbers that we know. 2 times Ï€ times the radius, the distance from the Sun to the Earth, 1.496 x 10^11 divided by the speed that it's traveling at, 29,779.3 meters per second.*1544

*Punch that into a calculator and what do we get? We get that it manages to make an orbit around the Sun in 3.156 x 10^7 seconds.*1566

*What does that mean? I don't know how much that is really, I'm not very good at knowing how many seconds is meaningful after a 100.*1582

*Let's figure out what that is for ourselves.*1593

*So how many seconds, what does that number of seconds mean in terms of minutes, in terms of hours, in terms of days?*1596

*Days would be good, we already know that the answer should be pretty close to 365 otherwise something has gone wrong, right?*1600

*We put that in and we have 3.156 x 10^7.*1607

*How many seconds are in a minute? 60. How many minutes are in an hour? 60.*1612

*How many hours are in a day? 24. Punch that into a calculator and we'll get 365.33 days.*1619

*Which is really good considering that the actual orbit of the Sun is a little bit less than 365 and quarter days.*1629

*Now you'll probably think that the orbit, that the Sun, that the Earth manages to get around the Sun every 365 days, that's not quite true.*1638

*365 days is the closest round numbers of day, but you know how we have leap years every four years?*1645

*The leap year every four years is to catch up with the fact that the Earth takes just a longer than a year to make it around the Sun.*1653

*So just because we talk a little bit longer than a year to make it around the Sun. We take one quarter of a day more.*1661

*We have to have 365 days, 365 days, 365 days, 366 days, 365 days and then in reality there is even more things that have to be corrected when you start to expand it.*1666

*It's actually 365 and a quarter minus just a little bit. So the fact that we have got 365.33 days when we simply this, we dealt with it as a circular orbit, which it's not perfectly.*1677

*We dealt with it as if the only force involved was the force of the Sun. The Sun's gravity on the Earth, which is not the case.*1687

*There is actually a bunch other things going on. We got a really, really good answer.*1693

*So just like with air resistance, you can manager to pretend it's not there sometimes and still get really good answers.*1699

*It's only when it's a really egregious thing to keep it out. When it's really bad, it's really important that it not be forgotten.*1703

*Like say, dropping a piece of paper, flat side down, that we're going to have to worry about the fact that we're getting rid of it.*1709

*In this case though, we really able to get a really close estimation.*1716

*Example 4, final example. A neutron star is a very dense type of star that rotates extremely quickly.*1720

*If a neutron star has a radius of 15 kilometers, which is the same thing as 1.5 x 10^4 meters, because we got to have things in standard meters.*1727

*It spins at a rate of 1 revolution per second, we can figure out what its minimum mass must be based on the fact that it doesn't fling itself apart.*1734

*What is that minimum mass?*1740

*We've got this thing spinning around very, very quickly. Say we consider some chuck of its surface, there's not really things on top of a neutron star.*1743

*The force of gravity so strong that it's going to just a pulp.*1751

*There is something, some chunk of it on the surface. Let's say that chunk has mass 1.*1755

*For it to continue to spin in a circle, because we're saying the neutron star is circular.*1761

*For it to be able to spin in a circle, it's got to have a centripetal force on it.*1765

*Force centripetal, what's that have to equal? Always has to be pointing in the center and equal the mass of the chunk times V squared over R.*1769

*V squared being actually the speed squared. Being a little bit lazy, sorry.*1780

*What forces are keeping it down? There is nothing holding it down, it's not tensioned to the surface.*1787

*The only thing holding it down is raw gravity. So we know that the force of gravity has to be at least equal to the force of centripetal.*1792

*We could have more than that right? Because there is pressure inside, so it could be larger gravity than that because just like you could have more gravity and still be attached to the Earth, you that the force of gravity has to be at least enough to hold up the centripetal force.*1799

*Then there could be a normal force to cancel out that extra gravity, but we know that the force of gravity...*1814

*has to be at least greater than or equal to the necessary centripetal force for that object to stay on the surface otherwise if the force of gravity is less than the centripetal force...pleh the entire neutron star will just explode out every which way and we won't have a neutron star anymore.*1821

*What will that minimum mass be? So let's look at the minimum case, which is going to be when the force of gravity is equal to the centripetal force.*1837

*If the force of gravity is equal to the centripetal force, we're going to get G M1 M2 over R squared equals that mass, that chunk on the surface, times V squared over R.*1845

*In this case, let's make that chunk on the surface M1 V squared over R.*1859

*M1's cancel out and we get G M2 over R squared equals V squared over R.*1864

*What are we looking for? Where looking for what the mass of the neutron star is, what the minimum mass is.*1872

*Remember we know force of gravity must be greater than or equal to, so the minimum mass is going to be when the force of gravity is equal to the centripetal force.*1879

*We want to solve for M2. M2 is going to equal to V squared.*1886

*We multiply both sides by R squared and that will leave us with an R up top and divided both sides by G, so we'll have G on the bottom.*1894

*Whatâ€™s V? Well, how fast is it moving around? If it makes one revolution per second, then that means how much distance it covers in that second.*1901

*So V is going to be equal to distance over time. Which is going to be equal to the circumference of the object divided by that one second because it manages to make one revolution in a second, right?*1911

*Circumference of the object is 2Ï€R divided by one second so we're left with just...*1925

*2Ï€R meters per second. So that's what the velocity is. We sub that in.*1933

*We get 2Ï€R squared times R over G or 4Ï€ squared distance the surface cubed divided by G.*1938

*We plug in a bunch of numbers, all those numbers that we have.*1956

*4Ï€ squared. What's R? The radius is 15 kilometers, so 1.5 x 10^4.*1960

*Now, it's not just squared now, it's not to the 1, it's cubed.*1972

*We divided this whole thing by G or 6.67 x 10^11. Sorry, not 10^11, 10^-11.*1977

*What do we get when we punch this all into the calculator? We're going to have to get some pretty large number right?*1988

*We've got 10^4 cubed up here times these other numbers and then divided by 10^11.*1992

*Since itâ€™s a negative exponent on the bottom it's going to wind up adding 10^11 on the top.*1999

*We punch that all through and the number that we wind up getting is 1.998 x 10^24 kilograms.*2003

*It has to be at least more than a third of the mass of the Earth otherwise it'll explode out in all directions.*2015

*In reality neutron stars turn out to be way more than that but we figured out what the minimum is based on the simple thing we've got right here.*2022

*The fact that it's rotating very quickly and it doesn't want to fling itself apart, it's got to have something holding itself in and we're assuming it's going to have to be holding itself in by gravity.*2029

*Because it's not a solid object, it's under that kind of gravity, it's kind of a soupy mass of things.*2041

*So to be able to keep itself from flinging itself apart it's got to have enough gravity to hold itself together.*2046

*Any object, any piece of itself, any mass of itself, must have enough force of gravity to overcome the necessary centripetal force.*2053

2 answers

Last reply by: Marcos Castillo

Sun Apr 13, 2014 12:57 PM

Post by Marcos Castillo on April 13, 2014

Hi, related to Newton's thought experiment, for the case we shoot the canon ball at the right velocity, in orden to describe an orbit, can we say that in this case gravity becomes centripetal acceleration?.

0 answers

Post by Ali barkhurdar on November 11, 2012

thanks

4 answers

Last reply by: Professor Selhorst-Jones

Tue May 28, 2013 4:22 PM

Post by ahmed raza on September 28, 2012

Is the formula F=G*M*m/r^2 related to newton's law of gravity which follows as "any two point masses attract each other with the force that is directly proportional to the product of their masses and inversely proportional to the square of their separation ", right?