WEBVTT physics/high-school-physics/selhorst-jones
00:00:00.000 --> 00:00:10.000
Hi. Welcome back to Educator.com. Today we’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.
00:00:10.000 --> 00:00:13.000
We’ll start to move on to other things in future sections, but this is it for mechanics.
00:00:13.000 --> 00:00:18.000
We got a good strong understanding of mechanics, you should be proud of yourself.
00:00:18.000 --> 00:00:26.000
So, basic introduction to gravity. What’s holding you to the Earth right now? What’s holding me to the Earth right now? Gravity, gravity is holding me down, keeping me on Earth.
00:00:26.000 --> 00:00:35.000
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.
00:00:35.000 --> 00:00:39.000
Gravity is one of the basic forces of the universe, really, really important.
00:00:39.000 --> 00:00:46.000
Gravity pulls massive objects together. The more mass you have, the more pull you exert on other massive objects you pull.
00:00:46.000 --> 00:00:54.000
Two objects, if they’re both really massive will pull together more than if one object has the same mass but the other one has a small mass.
00:00:54.000 --> 00:01:00.000
Also, the distance between them affects it. If you’re really close, you’ll wind up having more gravity than if you are very far away.
00:01:00.000 --> 00:01:08.000
Now, this idea of gravity, we’re used to it. We accept it right now, but keep in mind, the idea of gravity, was actually really once stridently fought against.
00:01:08.000 --> 00:01:22.000
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.
00:01:22.000 --> 00:01:28.000
The fact that humans are pulled on the same thing as on the stars seems ridiculous to some people, but it’s the true.
00:01:28.000 --> 00:01:31.000
Everything ends up having the same set of rules.
00:01:31.000 --> 00:01:41.000
Gravity, it’s actually a real thing and we’re used to that but keep in mind people didn’t always think that.
00:01:41.000 --> 00:01:45.000
Law of universal gravitation, formula for how gravity works.
00:01:45.000 --> 00:01:54.000
To derive a formula for the force of gravity, that’s kind beyond the scope of this course, so we’re just going to start by plucking it out of thin air.
00:01:54.000 --> 00:02:06.000
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.
00:02:06.000 --> 00:02:14.000
It’s an inverse square. If you’re farther away, it’s not just the distance divided, it’s the distance squared divided.
00:02:14.000 --> 00:02:23.000
Let’s talk about things more specific. Force of gravity, the size of force of gravity is equal to GxM1xM2/r2.
00:02:23.000 --> 00:02:29.000
M1 and M2, two mass of the objects involved, that’s pretty simple. R, is the distance between the two objects.
00:02:29.000 --> 00:02:35.000
If you really want to be specific, it’s more accurate to say it’s the center of the mass for each object.
00:02:35.000 --> 00:02:42.000
Now, keep in mind, we’re normally going to be dealing with very large distances and comparatively very small objects.
00:02:42.000 --> 00:02:50.000
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.
00:02:50.000 --> 00:03:02.000
We can worry about the center of mass but for the most part we’re going to be dealing with such large distances in any case, we don’t have to worry that much about the distance between them versus the distance between their centers of mass.
00:03:02.000 --> 00:03:05.000
Don’t worry about it too much, but keep in mind there is a slight difference there.
00:03:05.000 --> 00:03:11.000
G is the universal gravitational constant, which is the thing that makes this formula run.
00:03:11.000 --> 00:03:16.000
The idea is mass times mass divided by the square of the distance.
00:03:16.000 --> 00:03:27.000
That’s what affects it, but we need to have a specific thing that’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.
00:03:27.000 --> 00:03:34.000
6.67x10^-11. Newton’s times metered squared divided by kilograms squared.
00:03:34.000 --> 00:03:38.000
Because remember we want in the end to get Newton’s out of this.
00:03:38.000 --> 00:03:46.000
We want to get four sides of this, so if we got masses up top then we’re going to have kilograms squared up top.
00:03:46.000 --> 00:03:52.000
That will cancel out there. If we got on the bottom; meters. Then we got to cancel it out up top.
00:03:52.000 --> 00:04:01.000
We got meters squared times kilograms squared, so it will cancel out the two masses and dividing by a distance squared.
00:04:01.000 --> 00:04:09.000
That leaves us with Newton's. That's why the law gives us a unit that seems so bizarre.
00:04:09.000 --> 00:04:17.000
Now notice that G is a really tiny number. G is just incredibly small.
00:04:17.000 --> 00:04:23.000
The reason that we don't feel a pull from buildings around us is because G is so small.
00:04:23.000 --> 00:04:29.000
We're comparatively way closer to a building than the center of the Earth.
00:04:29.000 --> 00:04:37.000
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.
00:04:37.000 --> 00:04:41.000
It's a big number, somewhere on the scale of 10 to the 24th.
00:04:41.000 --> 00:04:47.000
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.
00:04:47.000 --> 00:04:57.000
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.
00:04:57.000 --> 00:05:03.000
Finally, force gravity is a vector. You have to remember it points between the two objects.
00:05:03.000 --> 00:05:12.000
Object 1, Object 2. Object 1 gets pulled towards Object 2, just like Object 2 gets pulled towards Object 1.
00:05:12.000 --> 00:05:19.000
Equal and opposite reactions; Newton's third law still applies. So the two objects pulled towards each other.
00:05:19.000 --> 00:05:25.000
Gravity is not just a number, it's a vector. You have to have a direction to go with that size.
00:05:25.000 --> 00:05:31.000
To go with that amount of force. Remember that it's always pulling towards the other object.
00:05:31.000 --> 00:05:39.000
Normally we'll be able to treat this as if it's single dimensional, but if you needed it would be actually vector quantity.
00:05:39.000 --> 00:05:44.000
So previously, we simply thought of gravity as a general acceleration.
00:05:44.000 --> 00:05:49.000
We knew G was equal to 9.8 meters per second per second.
00:05:49.000 --> 00:05:51.000
Now we're talking about universal gravitation. So what does that mean?
00:05:51.000 --> 00:05:56.000
What does that make our old conception of 9.8 per second per second into?
00:05:56.000 --> 00:05:59.000
Such an acceleration, we call a gravitation field.
00:05:59.000 --> 00:06:07.000
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.
00:06:07.000 --> 00:06:18.000
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.
00:06:18.000 --> 00:06:23.000
The world pretty much runs on that.
00:06:23.000 --> 00:06:25.000
How do we make these two come together?
00:06:25.000 --> 00:06:33.000
A gravitational field is a way of saying at a certain distance, you're going to experience a certain acceleration.
00:06:33.000 --> 00:06:37.000
How can we find gravitational fields in general?
00:06:37.000 --> 00:06:41.000
A gravitational field imposes a constant acceleration on anything inside of it.
00:06:41.000 --> 00:06:46.000
Remember before, we had force of gravity equal to the mass times the acceleration of gravity.
00:06:46.000 --> 00:06:50.000
Force of gravity equals the mass times the acceleration of gravity.
00:06:50.000 --> 00:06:54.000
For now, any object. This will work on Earth, but it will also work on anything.
00:06:54.000 --> 00:07:01.000
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.
00:07:01.000 --> 00:07:06.000
We could do it on the surface of the Sun, we could do with any object that we felt like.
00:07:06.000 --> 00:07:18.000
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.
00:07:18.000 --> 00:07:29.000
For example let’s talk about me. I will consider myself to be one of the masses.
00:07:29.000 --> 00:07:35.000
I'm Mass 1. I'm M1. I'm M1 times acceleration of gravity, is the force of gravity currently pulling on me.
00:07:35.000 --> 00:07:47.000
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.
00:07:47.000 --> 00:07:51.000
The distance between here and the center of the Earth.
00:07:51.000 --> 00:07:56.000
Mass times acceleration of gravity equals G times M1M2 over R squared.
00:07:56.000 --> 00:08:09.000
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.
00:08:09.000 --> 00:08:12.000
However far up we're putting our gravitational field.
00:08:12.000 --> 00:08:25.000
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.
00:08:25.000 --> 00:08:37.000
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.
00:08:37.000 --> 00:08:42.000
While the exact, the precise amount of gravity that is affecting me will change slightly.
00:08:42.000 --> 00:08:48.000
Its going to change a negligible amount. Which means that gravitational fields will work when we've got a very large, very massive object.
00:08:48.000 --> 00:08:57.000
The distance we're going to get from that object center point is very little compared to the distance of the whole thing.
00:08:57.000 --> 00:09:07.000
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.
00:09:07.000 --> 00:09:16.000
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.
00:09:16.000 --> 00:09:27.000
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.
00:09:27.000 --> 00:09:30.000
In orbit. Orbit is one body rotating around another.
00:09:30.000 --> 00:09:43.000
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.
00:09:43.000 --> 00:09:46.000
The force is equal to the mass times speed squared over the radius.
00:09:46.000 --> 00:09:56.000
So what is the centripetal force that keeps a celestial body rotating? That keeps celestial *bodies* rotating each other.
00:09:56.000 --> 00:10:01.000
What would that force be? Gravity.
00:10:01.000 --> 00:10:07.000
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.
00:10:07.000 --> 00:10:20.000
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.
00:10:20.000 --> 00:10:26.000
One thing to point out, this isn't just M. It's M1 or M2, depending on which one we want to make it.
00:10:26.000 --> 00:10:32.000
The object that's moving around, the M1's are going to cancel out on either side.
00:10:32.000 --> 00:10:38.000
The other thing to note is that I want to point out that in real life, orbits are almost never circular.
00:10:38.000 --> 00:10:43.000
Orbits can be close to circular but normally orbits are actually elliptical.
00:10:43.000 --> 00:10:48.000
A circle is something that has a constant radius. An ellipse is something that is able to squish out.
00:10:48.000 --> 00:11:03.000
An egg is kind of an ellipse. Things that get squish.
00:11:03.000 --> 00:11:10.000
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.
00:11:10.000 --> 00:11:16.000
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.
00:11:16.000 --> 00:11:22.000
Also, in real life, when the Earth is going around the Sun, there is something else working on it.
00:11:22.000 --> 00:11:27.000
All these other planets around us. Now comparatively the Sun is Big Pop in our universe.
00:11:27.000 --> 00:11:32.000
The Sun, it's got the most mass by far. It's able to have the most effect on our orbit.
00:11:32.000 --> 00:11:39.000
There is a whole but of other planets out there. One of the important planets that also has a really big mass, Jupiter.
00:11:39.000 --> 00:11:42.000
Jupiter has a really large mass compared to the mass of Earth.
00:11:42.000 --> 00:11:49.000
It's able to also have some effect on our orbit. Very little compared to the effect of the Sun.
00:11:49.000 --> 00:11:55.000
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.
00:11:55.000 --> 00:12:02.000
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.
00:12:02.000 --> 00:12:10.000
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.
00:12:10.000 --> 00:12:22.000
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.
00:12:22.000 --> 00:12:27.000
Except in really egregious cases where it's moving really fast, we have to clearly care about it.
00:12:27.000 --> 00:12:35.000
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.
00:12:35.000 --> 00:12:41.000
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.
00:12:41.000 --> 00:12:46.000
At the same, if wanted to be really rigorous, we would have deal other calculations and make it a whole lot harder.
00:12:46.000 --> 00:12:50.000
So like air resistance, we kind of put in on the table, left it for a later physics course.
00:12:50.000 --> 00:13:00.000
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.
00:13:00.000 --> 00:13:09.000
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.
00:13:09.000 --> 00:13:17.000
Imagine, and before I get too far, I would like to apologize for the bad drawing of this Earth, I am terrible at drawing.
00:13:17.000 --> 00:13:28.000
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.
00:13:28.000 --> 00:13:41.000
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.
00:13:41.000 --> 00:13:47.000
Now moving on. Imagine a very tall mountain on Earth. So tall as to be above the atmosphere.
00:13:47.000 --> 00:13:51.000
There will be no air resistance, so we don't have to worry about friction slowing down the object.
00:13:51.000 --> 00:13:57.000
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.
00:13:57.000 --> 00:14:00.000
What's going to happen as those velocities increase?
00:14:00.000 --> 00:14:07.000
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?
00:14:07.000 --> 00:14:14.000
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.
00:14:14.000 --> 00:14:19.000
The ball comes out and boom, falls right into the Earth. Well what if we put it would with a slight amount of force.
00:14:19.000 --> 00:14:24.000
Its going to shoot out, then it’s going to fall into the Earth.
00:14:24.000 --> 00:14:33.000
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?
00:14:33.000 --> 00:14:42.000
At every point on the circle, it's getting pulled in. Well if we shoot it harder, it's going to shoot out...
00:14:42.000 --> 00:14:56.000
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.
00:14:56.000 --> 00:15:03.000
It'll just go off forever. If it goes off forever, we've lost it, there is nothing there.
00:15:03.000 --> 00:15:12.000
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.
00:15:12.000 --> 00:15:21.000
If we shoot at the right speed, instead of falling into the Earth or falling out or away from the Earth.
00:15:21.000 --> 00:15:32.000
It's going to get pulled in and it's going to fall into the Earth forever and ever and ever.
00:15:32.000 --> 00:15:41.000
It's just going to keep spinning around the Earth because it's getting pulled in at all moments. So it just keeps going.
00:15:41.000 --> 00:15:50.000
The best one, the one that will be in orbit is a permanent fall. So the permanent fall is way to think of gravity.
00:15:50.000 --> 00:16:02.000
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.
00:16:02.000 --> 00:16:10.000
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.
00:16:10.000 --> 00:16:23.000
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.
00:16:23.000 --> 00:16:39.000
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.
00:16:39.000 --> 00:16:47.000
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.
00:16:47.000 --> 00:16:58.000
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.
00:16:58.000 --> 00:17:08.000
That's exactly what this is, it's a thought experiment that lets us understand a cool thing about the way the world works.
00:17:08.000 --> 00:17:16.000
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.
00:17:16.000 --> 00:17:21.000
If the centers of mass are 850 kilometers away from one another, what's the force of gravity between them?
00:17:21.000 --> 00:17:25.000
This is just a really blunt use of the force of gravity formula.
00:17:25.000 --> 00:17:35.000
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.
00:17:35.000 --> 00:17:40.000
Throw in all the numbers we have. We have 6.67 times 10 to the -11th.
00:17:40.000 --> 00:17:48.000
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.
00:17:48.000 --> 00:17:56.000
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.
00:17:56.000 --> 00:18:01.000
It's one of the basic fundamental concepts of the universe, so it matters.
00:18:01.000 --> 00:18:12.000
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.
00:18:12.000 --> 00:18:20.000
So mass object 1, mass object 2 divided by 850 kilometers, wait a second, standard units.
00:18:20.000 --> 00:18:24.000
What's the standard units here? Is kilometers standard? No.
00:18:24.000 --> 00:18:31.000
When we dealt with G, G required M squared. Remember it was M squared over kilometers squared times Newton’s.
00:18:31.000 --> 00:18:34.000
That M squared, we're going to have to be working in meters.
00:18:34.000 --> 00:18:42.000
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.
00:18:42.000 --> 00:18:55.000
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.
00:18:55.000 --> 00:19:00.000
Do the problem in normal metric units and then at the very end convert back to the unit that they gave you.
00:19:00.000 --> 00:19:09.000
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.
00:19:09.000 --> 00:19:17.000
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.
00:19:17.000 --> 00:19:22.000
To all of my viewers outside of America, you're probably not going to have to worry about other units.
00:19:22.000 --> 00:19:28.000
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.
00:19:28.000 --> 00:19:35.000
Anyway. Back to the problem. 6.67 times 10 to the -11th times, etc., divided by 850 kilometers, so what's that in meters.
00:19:35.000 --> 00:19:46.000
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.
00:19:46.000 --> 00:19:53.000
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?
00:19:53.000 --> 00:20:07.000
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.
00:20:07.000 --> 00:20:20.000
That's right, 900,000ths. 900,000ths of a Newton is how much they manage to pull on one another.
00:20:20.000 --> 00:20:24.000
These are fairly massive objects that are at distance that we wouldn't think is that huge.
00:20:24.000 --> 00:20:36.000
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 doesn’t have that much effect on us.
00:20:36.000 --> 00:20:45.000
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.
00:20:45.000 --> 00:20:48.000
What is the gravitational field on the surface of Earth?
00:20:48.000 --> 00:20:58.000
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.
00:20:58.000 --> 00:21:05.000
Force of gravity is equal to, we want to be something, mass times acceleration of gravity.
00:21:05.000 --> 00:21:13.000
We know force of gravity, we can change this into our general one. G M1 M2 over R squared equals mass.
00:21:13.000 --> 00:21:17.000
Lets make it mass 1 times acceleration of gravity.
00:21:17.000 --> 00:21:26.000
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.
00:21:26.000 --> 00:21:33.000
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.
00:21:33.000 --> 00:21:42.000
If that's the case, we got M1's cancel and we get G M2 over R squared equals the acceleration of gravity.
00:21:42.000 --> 00:22:14.000
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.
00:22:14.000 --> 00:22:20.000
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?
00:22:20.000 --> 00:22:32.000
Eventually it simplifies to 9.795 meters per second per second. Hey, that makes a lot of sense.
00:22:32.000 --> 00:22:36.000
What do we normal use? 9.8 meters per second per second.
00:22:36.000 --> 00:22:39.000
So this turns out to be something that works out well.
00:22:39.000 --> 00:22:42.000
Now there's a couple of simplifications that we made doing this problem.
00:22:42.000 --> 00:22:46.000
The Earth does not actually have the radius of this, because the Earth is not actually a circle.
00:22:46.000 --> 00:22:51.000
The Earth is slightly oblong, it's not quite a perfect circle.
00:22:51.000 --> 00:22:58.000
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.
00:22:58.000 --> 00:23:06.000
Also we don't necessarily have that the center of mass for the Earth is precisely in the center of the Earth.
00:23:06.000 --> 00:23:11.000
That might be the case, but we haven't been guarantee yet, we need to find out more about the composition of the Earth.
00:23:11.000 --> 00:23:14.000
So there's more things to keep in mind here.
00:23:14.000 --> 00:23:23.000
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.
00:23:23.000 --> 00:23:29.000
If the Sun has the mass of 1.9898 x 10^30 then that's also a big number.
00:23:29.000 --> 00:23:35.000
1.496 x 10^11 meters from the Earth, what velocity does Earth orbit the Sun at?
00:23:35.000 --> 00:23:39.000
How long does it take for the Earth to complete one orbit?
00:23:39.000 --> 00:23:44.000
We know that the force of gravity, because it's moving in a circle. Is there any other forces operating on it?
00:23:44.000 --> 00:23:52.000
No, we know that it's just centripetal force pulling. Just gravity pulling, so that must be the entirety of our centripetal force.
00:23:52.000 --> 00:24:00.000
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.
00:24:00.000 --> 00:24:10.000
G times M1 M2 over R squared equals, what's the centripetal force, M1 V squared over R.
00:24:10.000 --> 00:24:15.000
M1 in this case is the object moving around. It can be canceled.
00:24:15.000 --> 00:24:24.000
We've got that G M2 divided by, let's multiply both sides by R, equals V squared.
00:24:24.000 --> 00:24:29.000
Now we'll take the square root and we'll get G M2 over R equals V.
00:24:29.000 --> 00:24:47.000
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.
00:24:47.000 --> 00:24:55.000
All divided by the distance between the Earth and the Sun square, sorry not squared, because we managed to cancel out those R's.
00:24:55.000 --> 00:25:04.000
1.496 x 10^11th.
00:25:04.000 --> 00:25:16.000
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.
00:25:16.000 --> 00:25:19.000
The Earth is really whizzing through the solar system.
00:25:19.000 --> 00:25:25.000
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?
00:25:25.000 --> 00:25:32.000
Well the circumference, the path it has to follow is 2πR right?
00:25:32.000 --> 00:25:44.000
So the orbit time is going to be, how far a distance it has to go, 2πR divided by the speed that it's moving at, V.
00:25:44.000 --> 00:26:06.000
We start substituting in the numbers that we know. 2 times π 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.
00:26:06.000 --> 00:26:22.000
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.
00:26:22.000 --> 00:26:33.000
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.
00:26:33.000 --> 00:26:36.000
Let's figure out what that is for ourselves.
00:26:36.000 --> 00:26:40.000
So how many seconds, what does that number of seconds mean in terms of minutes, in terms of hours, in terms of days?
00:26:40.000 --> 00:26:47.000
Days would be good, we already know that the answer should be pretty close to 365 otherwise something has gone wrong, right?
00:26:47.000 --> 00:26:52.000
We put that in and we have 3.156 x 10^7.
00:26:52.000 --> 00:26:59.000
How many seconds are in a minute? 60. How many minutes are in an hour? 60.
00:26:59.000 --> 00:27:09.000
How many hours are in a day? 24. Punch that into a calculator and we'll get 365.33 days.
00:27:09.000 --> 00:27:18.000
Which is really good considering that the actual orbit of the Sun is a little bit less than 365 and quarter days.
00:27:18.000 --> 00:27:25.000
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.
00:27:25.000 --> 00:27:33.000
365 days is the closest round numbers of day, but you know how we have leap years every four years?
00:27:33.000 --> 00:27:41.000
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.
00:27:41.000 --> 00:27:46.000
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.
00:27:46.000 --> 00:27:57.000
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.
00:27:57.000 --> 00:28:07.000
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.
00:28:07.000 --> 00:28:13.000
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.
00:28:13.000 --> 00:28:19.000
There is actually a bunch other things going on. We got a really, really good answer.
00:28:19.000 --> 00:28:23.000
So just like with air resistance, you can manager to pretend it's not there sometimes and still get really good answers.
00:28:23.000 --> 00:28:29.000
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.
00:28:29.000 --> 00:28:36.000
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.
00:28:36.000 --> 00:28:40.000
In this case though, we really able to get a really close estimation.
00:28:40.000 --> 00:28:47.000
Example 4, final example. A neutron star is a very dense type of star that rotates extremely quickly.
00:28:47.000 --> 00:28:54.000
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.
00:28:54.000 --> 00:29:00.000
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.
00:29:00.000 --> 00:29:03.000
What is that minimum mass?
00:29:03.000 --> 00:29:11.000
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.
00:29:11.000 --> 00:29:15.000
The force of gravity so strong that it's going to just a pulp.
00:29:15.000 --> 00:29:21.000
There is something, some chunk of it on the surface. Let's say that chunk has mass 1.
00:29:21.000 --> 00:29:25.000
For it to continue to spin in a circle, because we're saying the neutron star is circular.
00:29:25.000 --> 00:29:29.000
For it to be able to spin in a circle, it's got to have a centripetal force on it.
00:29:29.000 --> 00:29:40.000
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.
00:29:40.000 --> 00:29:47.000
V squared being actually the speed squared. Being a little bit lazy, sorry.
00:29:47.000 --> 00:29:52.000
What forces are keeping it down? There is nothing holding it down, it's not tensioned to the surface.
00:29:52.000 --> 00:29:59.000
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.
00:29:59.000 --> 00:30:14.000
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.
00:30:14.000 --> 00:30:21.000
Then there could be a normal force to cancel out that extra gravity, but we know that the force of gravity...
00:30:21.000 --> 00:30:37.000
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.
00:30:37.000 --> 00:30:45.000
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.
00:30:45.000 --> 00:30:59.000
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.
00:30:59.000 --> 00:31:04.000
In this case, let's make that chunk on the surface M1 V squared over R.
00:31:04.000 --> 00:31:12.000
M1's cancel out and we get G M2 over R squared equals V squared over R.
00:31:12.000 --> 00:31:19.000
What are we looking for? Where looking for what the mass of the neutron star is, what the minimum mass is.
00:31:19.000 --> 00:31:26.000
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.
00:31:26.000 --> 00:31:34.000
We want to solve for M2. M2 is going to equal to V squared.
00:31:34.000 --> 00:31:41.000
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.
00:31:41.000 --> 00:31:51.000
What’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.
00:31:51.000 --> 00:32:05.000
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?
00:32:05.000 --> 00:32:13.000
Circumference of the object is 2πR divided by one second so we're left with just...
00:32:13.000 --> 00:32:18.000
2πR meters per second. So that's what the velocity is. We sub that in.
00:32:18.000 --> 00:32:36.000
We get 2πR squared times R over G or 4π squared distance the surface cubed divided by G.
00:32:36.000 --> 00:32:40.000
We plug in a bunch of numbers, all those numbers that we have.
00:32:40.000 --> 00:32:52.000
4π squared. What's R? The radius is 15 kilometers, so 1.5 x 10^4.
00:32:52.000 --> 00:32:57.000
Now, it's not just squared now, it's not to the 1, it's cubed.
00:32:57.000 --> 00:33:08.000
We divided this whole thing by G or 6.67 x 10^11. Sorry, not 10^11, 10^-11.
00:33:08.000 --> 00:33:12.000
What do we get when we punch this all into the calculator? We're going to have to get some pretty large number right?
00:33:12.000 --> 00:33:19.000
We've got 10^4 cubed up here times these other numbers and then divided by 10^11.
00:33:19.000 --> 00:33:23.000
Since it’s a negative exponent on the bottom it's going to wind up adding 10^11 on the top.
00:33:23.000 --> 00:33:35.000
We punch that all through and the number that we wind up getting is 1.998 x 10^24 kilograms.
00:33:35.000 --> 00:33:42.000
It has to be at least more than a third of the mass of the Earth otherwise it'll explode out in all directions.
00:33:42.000 --> 00:33:49.000
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.
00:33:49.000 --> 00:34:01.000
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.
00:34:01.000 --> 00:34:06.000
Because it's not a solid object, it's under that kind of gravity, it's kind of a soupy mass of things.
00:34:06.000 --> 00:34:13.000
So to be able to keep itself from flinging itself apart it's got to have enough gravity to hold itself together.
00:34:13.000 --> 00:34:53.000
Any object, any piece of itself, any mass of itself, must have enough force of gravity to overcome the necessary centripetal force.