(host) This is a candle inside of a completely closed box. We're about to drop it 11 meters and see what happens to the flame. Check it out--it becomes completely spherical. This maybe isn't surprising if you've ever seen pictures of candles in space. They become this eerie, spherical shape. But wait, why should a candle in space and a candle falling in a box do the same thing? We're going to figure that out today. Hi, I'm Dianna Cowern, and welcome to lesson 8 of Dianna's intro physics class, also known as AP Physics 1 Review, also known as Physics by Dianna. Today's lesson is on gravity and orbits. And so we need to take a tour through space.

(Describer) She puts. on a hoodie with a galaxy design and a face shield.

Check it out. Grab something nearby and hold it in your hand. I just so happen to have an apple, the object most closely associated with gravity in physics. I can feel it being pulled toward the ground. Isn't that kind of weird? I mean, there's no string here. There's nothing underneath, but the object is totally being pulled toward the ground. Why? Why? What is going on?

(Describer) She drops it and gets a pineapple.

If I hold up an even heavier apple, I can feel even more force. Why is that? Well, I'm going to tell you something amazing. Our answer to this question also explains how planets orbit the sun in our solar system and, even further, lets us find extrasolar planets orbiting stars throughout our galaxy, outside of our solar system. So by the end of today's lesson, this is what you're going to understand. You're going to understand how gravity, which is a force that makes an apple fall from a tree; it's also a force that makes stars, planets, moons, and even galaxies orbit one another. It's the same force, and we're going to use our new physics to describe, mathematically, what those orbits look like--crazy. So got everything you need? Snacks, calculator, space suit. The big idea today is Newton's law of universal gravitation. Let's go. This is Earth. Guess what all of those dots are, though? I'll give you a hint. They are not flies. These are all satellites. Yeah, fun fact, there are over 2,000 satellites up there. If you think that's a lot, guess how many stars are orbiting the supermassive black hole at the center of our galaxy. I'll give you a hint-- it's 6,000 times the number of flies on Earth. That's right--it is 100,000 million stars. All this orbiting happens because of gravity. So let's explore the history of how we even discovered gravity, how we even got the idea of gravity. Imagine a time before we knew gravity existed, before we knew what held our planet, Earth, in orbit around the sun. Astronomers were perplexed. In the late 1500s, they didn't know about the orbit yet. They were just tracking the motions of the planets and the sun really carefully and being like, hmm, there are patterns, curious. And then eventually, one dude suggested that Earth was orbiting the sun, which, girl, back then was a blasphemous idea. That was Copernicus in the '40s-- the 1540s. And it was a clear explanation for the patterns of these motions. And then another couple dudes, Johannes Kepler and Tycho Brahe, in the early 1600s, watched how all the planets moved and came up with a few laws to describe the planetary motion, but they still didn't have gravity. And then in 1610, Galileo used one of the first telescopes to look at Jupiter and saw-- holy Europa! Jupiter has moons too. So by the mid-17th century, all these astronomers had a good understanding of how all these planets and stuff moved, but not why. Remember, with no gravity, it didn't make a whole lot of sense that Earth would orbit the sun or that Jupiter's moons would stick around. That must have been baffling-- until Isaac Newton came along. Newton's big idea: gravity. There's this attractive force between all objects with mass-- moons and planets, planets and suns, apples and Earth. Yes, even between two fruit, there is a gravitational attraction. And Newton gave us a mathematical model of it.

(Describer) She catches a thrown marker.

The gravitational force between two objects, F, is proportional to their masses, M and M,

(Describer) Capital M and small m.

and inversely proportional to the distance between their masses, r squared.

(Describer) She writes on paper.

r is not the distance between two objects; it's the distance between their centers of masses, which, for most planets, is pretty much right in the center. And the constant of proportionality-- that's just the number that we multiply everything by to make it all work-- we call it big G. Look, this equation is so important it made it

(Describer) The equation hangs behind her.

into the air on our set...

(Describer) (no standard option) F equals G-M-m over r-squared.

(describer) F equals GMm over r squared. Big G. Now if you and I use this equation and we play a little bit, we can get some intuition for what it means. Let's try making either of these masses bigger. So if we make one of these masses bigger, what happens to the force? Well, it goes up. That means the force of gravity between me and Earth is bigger than the force of gravity between a pineapple and the Earth. And that force gets smaller as you move away from the center of the Earth. So if your r increases, then your force would decrease. In case you didn't know, you're not at the center of the Earth. You're standing a distance away from it, specifically 6,400 kilometers from the center of the Earth. So if you could move up out of the atmosphere another 6,400 kilometers, up to here, then you are doubling your distance. So put two up here. Looks like a man with a really big barbell. Imagine it's still you, up at twice the distance. So now you've doubled your distance to the center of the Earth. So you're doubling your r. So now you're dividing by a bigger r squared, and our model tells us that that would decrease the force of gravity between you and the Earth by a factor of 4, because you're squaring the 2. That's what our mathematical model tells us. And guess what? It's true. Distance matters. A great example of this would be ocean tides. If the moon disappeared, what would happen to the tides on Earth? Would they go away? Actually, no--they would get smaller. But there's something that would keep them going. It's the sun's gravitational pull, which also contributes to the tides, but even though the sun is much bigger, the moon is a lot closer. So r matters. All right, now, big G-- this is what we call the universal gravitational constant, and it's a number that we just measure. And as far as we know, it's constant throughout our universe. Is it different in some other universe? Could be, we don't know. But in our universe, it is 6.67 times 10 to the minus 11. Let's do the units. On the left, we got newtons for our force equals, and then whatever the units are of G, times M is the kilograms. These are both kilograms. So we got kilograms squared over... and then r squared is going to give us meters squared. So if we rearrange that, we get newtons times meters squared over kilograms squared. And that's my units of G. So that means our units are newton meters squared per kilogram squared, and that is G. So thank you to all of those other astronomers for giving us the lead up to this. And then Newton gave us this tool right here for understanding gravitation. So let's use it. Let's start right here on Earth. What's the force of gravity on some object-- say, a falling pineapple on Earth? So here's my force of gravity, F of G, and we're going to say the mass is 1 kilogram. So, simple--we know that the force of gravity is going to be the gravitational constant. So F of G is the gravitational constant times the two masses divided by the distance between them. The gravitational constant is 6.67 times 10 to the minus 11. Units are newton meters squared per kilogram squared. And then the mass of the Earth, got to look it up, but it's 5.97 times 10 to the 24 kilograms. That's a big old planet. And then the mass of our pineapple, 1 kilogram, over, and then the distance between them. So that's roughly the radius of the Earth. So the radius of the Earth is about 6,370 kilometers, which is 6.37 times 10 to the 6 meters, because we want to work in meters per SI unit. We're going to square that. And don't forget to always work in meters so that the units match. And if I plug in all of these numbers and do the math, I get that the force of gravity on my 1-kilogram pineapple is 9.81 newtons. That's the force between this pineapple and the Earth. And remember, the force is a two-way street. So the Earth is pulling on the pineapple, and the pineapple is also pulling on the Earth, but the Earth doesn't really feel it. 9.81, that number looks familiar. We already know that objects near Earth are accelerating at 9.81 meters per second squared. So how did this pop out? Well, it's because F equals ma.

(Describer) She writes that.

And the force of gravity on a mass near Earth is GMm over r squared. So check it out--the mass of the object cancels out. And so then the acceleration only depends on G, big M, and r. So this is another way to show you that whether you drop a fish on a fishing pole or a Slinky or an entire water park from a tall building, the objects will all accelerate toward the ground at the exact same rate. And if you plug in the mass of Earth and the radius of Earth here, and G, out pops a equals 9.81 meters per second squared. By the way, it's the center of mass of an object that accelerates at 9.81 meters per second squared. Check out this really cool video of my friend Derek from the YouTube channel Veritasium dropping a Slinky in super slow motion. Look at the bottom of the Slinky. It almost doesn't move. It looks like it's defying gravity. But the Slinky is not a rigid object. It's collapsing, so the center of mass of it is accelerating down at 9.81 meters per second squared as the Slinky changes shape. That looks awesome. But what does all this stuff about Slinkys and accelerations have to do with orbits? Here's what. The force of gravity that accelerates pineapples down toward the Earth is the same force that binds the Earth to the sun. This is a big idea. The law of gravitation applies across huge expanses of space and explains why massive planets like Jupiter stay in their orbit around the sun, even though they're really, really, really far away. Let's analyze an orbit. It's a pretty neat idea. And, actually, the more you think about it, the more confusing it gets. Think about the International Space Station. So it's up there at about 400 kilometers above Earth's surface. So let's draw that-- actually 408 kilometers above Earth's surface. If we had just put it there, it would fall straight down. But if we gave it a sideways velocity, well, it might fall-- it might fall in that parabolic shape that we learned about in ballistic motion episode. If we give it a massive, huge sideways velocity, then it might fly off of Earth and out into space and never come back. But a just right amount of huge sideways velocity, and it'll fall around Earth and keep missing. And that's how you get orbit. An object in orbit is in freefall. We saw evidence for this with the candle in the box. The candle looks just like a candle orbiting the Earth on the Space Station. They're both in freefall. The only difference is that one is moving sideways fast, and the other is not. So the ISS is falling around Earth, and that orbital motion could be circular, like I drew. But orbits don't have to be circular. Comets, they orbit in very eccentric ellipses, but most planets and most moons orbit in ellipses that are pretty close to circles. So, hey, circular motion, we have a tool for that. The sum of the forces on an object moving in a circle will be mv squared over r. And what is this force? Well, it's the gravitational force between the ISS and the Earth. So that, GMm over r squared. Look, the masses cancel again, and we get an equation relating the sideways speed that you need to orbit a big old planet with mass M at a distance r from its center. We've got to v squared. We're going to cancel this r with one of these, and we're going to have to square root to get rid of that. So we'll end up getting v equals the square root of GM over r. So this is our equation for orbital velocity, based off of the mass and the radius of the thing that you're orbiting. The masses cancel. So the mass of the ISS doesn't affect its orbital velocity. It better not, otherwise when astronauts stepped outside to do some repairs, they would have a real problem. So what is the sideways velocity of the ISS? Let's work it out, now that we have the equation. Its height averages about 410 kilometers above the surface of the Earth, so the mass of Earth is about 5.97 times 10 to the 24 kilograms. And since the ISS orbits at about 410 kilometers above the surface of the Earth, the radius of the Earth is about 6,370 kilometers. Altogether, this is about 410 plus 6,370 is 6,780 kilometers. But again, we want to work in meters, so that's about 6.78 times 10 to the 6th meters. And then that whole thing is square-rooted. And when we do all this math, we end up getting about 7,660 meters per second. And that is the orbital velocity of the International Space Station around Earth. That's pretty cool. If you actually go and search on the internet for the orbital velocity of the ISS, you get 7,660 meters per second, which is awesome. But we just worked it out from the math, and you get the same value as you would search, and that it actually is in reality. That's really fast-- 7,000, almost 8,000 meters per second, 8 times 10 to the 3rd meters per second-- but not quite as fast as the speed of gravitational waves. I'm going to go off in a little tangent here. So if something violent happens, like two black holes collide, you can get a gravitational wave, this ripple in space moving across the universe. And that moves at the speed about 3 times 10 to the 8th meters per second, which might sound like the speed of light to you. And in fact, gravitational waves do move at the speed of light. So that's 10 to the 5th times the speed, the orbital speed of the ISS, so very fast. But this is impressive also. OK, let's move on. So now you have a tool that can tell you how fast do you need to go in order to orbit a mass at a particular distance from its center mass. Here's the thing. It's kind of hard to tell how fast you're going in space. We don't exactly have mile markers. So what if we can rewrite our equation to tell us something more useful, specifically the time that it takes you to orbit? So start with our orbital velocity equation, and we'll write it like this. We're going to just square that and get v squared equals GM over R. And then recall that we're orbiting in a circle, which means that the distance in one complete revolution-- you know it--it's equal to the circumference of the orbit, or 2 pi times the radius, r. And it's easy to measure the time it takes to complete one orbit. Then we'll call that orbital period, the time, big T; so that's our orbital period. So the velocity turns out to be easy to measure in terms of the distance traveled in one orbit, 2 pi r, and the orbital period, T. So our velocity is delta x over delta T, which is 2 pi times the radius of our orbit over our orbital period. So now let's plug it in. So we've got 2 pi r over T squared. And then that equals GM over r. So then we can rewrite this as 4 pi squared, r squared, over T squared. Rearrange this a little bit, and we get 4 pi squared, r cubed, equals GMT squared. Boom, GMT squared equals 4 pi squared r cubed. You may have seen something like this before, stated as Kepler's Third Law. And in fact, Kepler did observe this relationship empirically, in the sky. And then Newton showed us how it worked. This is a fantastic tool. This is the mass measuring tool that I promised. Check take it out--if you can observe how long it takes a planet to orbit the sun, for example, and if you can find the radius of the planet's orbit-- that's r--so that would be the distance from-- that would be approximately the distance from the sun to the Earth, then you can calculate the mass of the sun. You can measure the mass of the sun. So let's do it! The Earth's orbital radius, r, is about 150 million kilometers, so that's 1.5 times 10 to the 11 meters. We've got 4 pi squared times 1.5, 1.5 times 10 to the 11 meters cubed. G equals 6.67 times 10 to the minus 11. Newton meters squared per kilogram squared, and then we've got M, and then we've got our orbital period, which is 365 and 1/4 days, which we want in seconds. So that worked out to be about 31.6 million seconds, so that's 3.16 times 10 to the 7 seconds squared. Ooh, big numbers, and all of this works out to give you 2 times 10 to the 30th kilograms. That is a lot of mass. In terms of Earth masses, that's about 330,000 Earths' worth of mass. We just weighed the sun. That's so cool. I bet you didn't put down your morning to-do list. Astronomy is amazing. There is so much we just did and that astronomers have done with just a few different tools. Just like we did, they determine the mass of the supermassive black hole at the center of our galaxy using Kepler's Third Law on the stars orbiting nearby the supermassive black hole. And my favorite use of these tools-- astronomers realize that there's mass out there in the universe that we can't see. It's called dark matter. In the '60s and '70s, an astronomer named Vera Rubin measured how fast stars in spiral galaxies were orbiting around their centers, orbiting around the center of the galaxies. And she found that the stars on the edge were going much faster than they would have expected based off all those stars and things that we could see, which meant that there had to be tons of mass in these galaxies that wasn't emitting or reflecting any light. We called it dark matter. And since then, astronomers have found lots and lots of evidence for dark matter, and they've realized that it's super common. In fact, it makes up about 23% of the mass in our universe. And we still have no idea what it is. We've ruled out, like, dark planets and dead stars. We don't know what dark matter is yet. And we learned all of that from this, from GMT squared equals 4 pi squared r cubed. But there's more. I want to give you a quick peek at some really mind-boggling physics from the theory of general relativity, which is Einstein's theory that gravity can actually bend light. So when I first heard about Einstein's theory of relativity, I was like, wait, so all of that stuff I learned about Newton's laws, about gravity, is wrong? And no, that's not quite true. It's just that Newton's model is not quite as precise as Einstein's model. But Newton's model is really helpful in most situations we're dealing with. What Einstein's more precise model says is this-- in an earlier lesson, we said that the mass is the measure of the inertia of an object. So for example, this cow-- the mass tells us how hard we need to push it to accelerate it, F equals ma. So that's the inertial-- that's what we call the inertial mass of an object. And now we've seen that the mass of an object also determines how much gravitational attraction it will be between it and other objects. So F equals GMm over r squared. The M in there is the object's gravitational mass. And when we derived little g earlier, we set those two forces equal each other. And I did this in my physics class, and people have done it for centuries, and we canceled out the m's. But think about it for a minute-- it works, but should we be able to just do that? Why are inertial and gravitational mass the same thing? Why is the m that determines inertia the same as the m that determines what gravity it feels? By the way, I didn't wonder this until after I finished my degree in physics. It's kind of a mind-blowing question. But enter Einstein. One of his many huge contributions to physics was an idea called the equivalence principle. He stated that the effects of a gravitational field and the effects of accelerating motion are identical and cannot be distinguished. Whether you're sitting in your bedroom on Earth in a gravitational field or whether aliens have kidnapped you and put you on a spaceship accelerating at 9.8 meters per second squared, you would not know. You wouldn't notice any differences in how apples fall or how heavy you feel, because in a spaceship, your apple isn't actually falling, because there's no large gravitational body around to pull the apple. Imagine instead of dropping an apple, you shoot a laser beam across your bedroom in the alien spaceship. The laser beam shoots in a straight line through space, but like in the apple example, the spaceship is accelerating up into the laser beam. So by the time the beam has reached the other side of the room, the spaceship has moved, and the beam will hit a slightly lower part of the wall. By the equivalence principle, the same thing would have to happen to a light beam in a gravitational field. The light beam would have to bend toward the massive object generating that field, and that leads to the idea that massive objects are actually bending spacetime around them. So light beams traveling through spacetime sometimes end up on curved and bent paths. How about them apples? So that's today's lesson on gravitation and orbits.

(Describer) She moves the paper.

When they ask you what you learned on YouTube today, here are your two key takeaways. One, Newton's laws of gravitation. All masses attract each other with a force of F equals GMm over r squared. And two, orbital periods-- the square of an orbital period is proportional to the mass of a thing being orbited and the cube of the orbital radius. So 4 pi squared over T squared equals GM over r cubed. Here's a recap of all the problems we did because the best way to make sure that you get these concepts is to work the problems yourself. And for the problems we did in this lesson, like calculating the acceleration of gravity on Earth or finding the orbital velocity of the ISS, you can easily give yourself more practice by repeating them for other solar-system planets.

(describer) Problem one: An object with mass M

(Describer) Problem 1: An object with mass M is a distance R from the center of the Earth. Another object with mass 2M is at a distance 3R from the center of the Earth. What is the ratio of the forces between each object and the Earth? Problem 2: What is the force of gravity between the Earth and a pineapple with a mass of 1 kilogram, if the pineapple is falling near the Earth's surface? Problem 3: What would the acceleration due to gravity be on a planet with half the radius, but twice the mass of Earth? Problem 4: The ISS orbits with an average altitude of 408 kilometers. What is its orbital velocity? What is its orbital period? Extra Credit: What is the orbital velocity of an object orbiting the Earth at a height of 35,800 kilometers above the Earth's equator? (Don't forget to add in the Earth's radius!) What is the orbital period?

is a distance R from the center of the Earth. Another object with mass 2M is at a distance 3R from the center of the Earth. What is the ratio of the forces between each object and the Earth? Problem two: What is the force of gravity between the Earth and a pineapple with mass of 1 kilogram, if the pineapple is falling near the Earth's surface? Problem three: What would the accelertion due to gravity be on a planet with half the radius but twice the mass of Earth? Problem four: The ISS orbits with an average altitude of 408 kilometers. What is its orbital velocity? What is its orbital period? Extra credit: What is the orbital velocity of an object orbiting the Earth at a height of 35,800 kilometers above the Earth's equator? (Don't forget to add in the Earth's radius!) What is the orbital period? Also, we focused today on the solar system, but the last bit of science I want to share is about exoplanets. There are other planets outside of the solar system that we have found, like Kepler-78b, which is a planet that orbits extremely close to its star, at only 900,000 kilometers. And for comparison our closest planet, Mercury, which is our closest planet, the closest it gets the sun is 45.9 million kilometers. So Kepler-78b is really close to its star. Its orbital period is only 8 and 1/2 hours. And because it's so close, the surface temperatures reach 2,400 degrees Celsius, which is 9 times as hot as Venus. But in the next 3 billion years, the star's gravity will eventually suck it in and will completely consume it. So there's that. We also look at the orbital periods of exoplanets to determine the radius of their orbits. And from that, we find planets that live in the Goldilocks zone for life, which is where water will be in the liquid phase. So you get to learn a whole lot about exoplanets and more if you continue on with physics and take a class in astrophysics. I do hope you continue on with physics. And now, here's a message for you from a special guest. Hi, everyone. Congratulations on finishing another amazing lesson put together by the wonderful Physics Girl, Dianna. So I just want to say, stick with it because once you understand physics, it's like you have this whole superpower to see so many things in the world. I just want to say, though, on your journey, it's going to be really, really intense because, oh my gosh, there's going to be new equations, these concepts, a bunch people's names associated with different laws of whatever. So you know what? Just ask questions. It's going to be really helpful for you, it'll be helpful for other people. Ask them in forums, find different friends to talk to, watch different videos-- like, just keep at it, because it's great. So stick with it. You got this. I'm Dr. Brittany Kamai. I'm an astrophysicist. I work on building instrumentation to study the universe, specifically gravitational wave detectors, looking at these ripples in spacetime. And I think about how to make these detectors even better by using this other cool thing that's on the scene, which is metamaterials. These are people-designed materials defying what we thought of physics before. So good luck, put in the work. Can't wait to see what comes out of it afterwards. Take care.

(Describer) Titles: Executive Producer/Host/Writer: Dianna Cowern Lead Writer/Course Designer: Jeff Brock Producer: Laura Chernikoff Video Editor: Victoria C. Page Videographer: Levi Butner Curriculum Consultants: Lucy Brock, Samantha Ward Consulting Producer: Vanessa Hill Curriculum Consultant: Andy Brown Physics Girl Accessibility provided by the US Department of Education.

Accessibility provided by the U.S. Department of Education.