Everyone. So in this video, we're going to talk about some fascinating objects in our universe which are called black holes. All right. So there's one important equation that you need to know to solve problems. So let's go and take a look and we'll do a quick example. All right. So a black hole is an object that has an enormous amount of mass in a relatively tiny space. And what I mean by that is that something like 500 kilometers is pretty big to us, but relative to the solar system and it's incredibly small distance. All right. So it's relatively tiny. Now, this object is so massive that not even light can escape. So basically anything that falls into a black hole, whether it's a planet or a, or a star or something like that or a spaceship or even light itself once it falls in and it can never come back out. And the reason for that is if you've seen escape velocity, OK. If you haven't, basically, the escape speed would have to be faster than the speed of light, which is three times 10 to the 8 m per second. But nothing can go faster than light. So, essentially anything that falls in is just a doomed and you'll just fall in, you'll never be able to come back out. And that's why it looks black to us is because no light escapes it and actually reaches our eyes or our telescopes and stuff like that. All right. So there's an important equation called the schwarzschild radius. And it's the equation that relates the mass of the black hole and the physical size of the black hole. The way that we define the size of the black hole is basically the radius of the center out to this boundary here where you see all the lights and that's called the Schwarzschild radius. The equation is actually pretty straightforward. It's two G and then we're gonna do MBH for black hole divided by C, all right. And so basically the surface of this boundary here, the boundary where everything gets sort of dark is called the event horizon. So this boundary here is called the event horizon. It's more of a mathematical boundary. It's not like if you were to sort of pass through it and feel like a barrier or something like that. But basically what happens is that once you pass or cross this boundary, you'd have to go faster than the speed of light to come back out again. But that's impossible. So this is the boundary where nothing can possibly escape and you're doomed if you fall in. All right, So everything else that we learned in this chapter, all the stuff about forces satellite motion, all of our equations are still valid. We just now have an extra one, the Schwartzchild radius equation. All right. So let's go ahead and take a look at our example here. So we have a team of astronomers who are imaging a black hole at the center of our galaxy or the galaxy MA DC. But by the way, this is a real thing that happened and what they determined is anything closer than this distance here. 100 and 20 A U falls in and never escapes. Now, you should recognize that as the, basically the distance in which uh something comes in to the black hole and then never comes back out. So this is gonna be the short shell radius. We wanna calculate the mass of this black hole. But we want, in terms of solar masses in terms of basically as a multiple of how much ma how massive our sun is. OK. So we want the MBH, but we want it in terms of M suns, right? So if we want the mass of the black hole, we're gonna have to use our new equation, the, the Schwartzchild radius. So this is RS equals two G mass black hole divided by C squared. We want this MBH, we're just gonna have to rearrange, the C squared goes up to the top, the two G comes down and you're gonna have RSC squared times C squared divided by two G and this is gonna equal the mass of your black hole. All right. So we know what the schwarzschild radius is. It's the 120 A U. But because we're calculating a mass, we need everything to be in SI units. So this RS here, which is 100 and 20 A U, we're gonna have to convert it to meters and I have this conversion factor right here. So if you want it in terms of meters, you're gonna have to cancel the A U on the bottom. So this is gonna be 1.5 times 10 to the 11th meters and that's gonna get rid of uh and, and, and then that's gonna get rid of your A US. All right. So this just becomes 1.38 uh this 0.0 sorry, 1.8 times 10 to the 13 m. Ok? So now this is gonna be 1.8 times 10 to the 13 and then you're gonna do three times 10 to the eighth. That's the speed of light, but you have to square it and now you divide by two times 6.67 times 10 to the minus 11. That's your big G uh minus 11. And then when you work this out, what you're gonna get here is you're gonna get a number that's 1.21 times 10 of the 40 kg. All right. Now, this might not seem so you know, uh might seem like any ordinary big number. But remember we wanna, we wanna express this in terms of solar masses. OK? So there's one last conversion I need to do, which is I want to convert this kilograms here to M. So, all right, so what I have to do is I'm gonna have to multiply this so that the kilograms cancels on the bottom. So this is gonna be two times 10 to the 30. This is gonna be one. this is gonna be M sun. So when you work this out, your kilogram should cancel and the mass of your black hole is gonna be six times 10 to the ninth M sun. So if you uh so for those of you who uh realize what this number is, this is gonna be about 6 billion. So 6 billion times the mass of our sun. So in other words, this black hole is 6 billion times heavier than our sun. All right. And it's in a relatively small space, 100 and 20 A U isn't a very, very big space in, in the terms of like a galaxy or something like that. This is basically like the size of our solar system. All right. So again, very, very enormous mass, very sort of relatively tiny amount of space. That's it. For this one guys. Thanks for watching. Let me know if you have any questions.
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Problem
Problem
Astronomers have found a small, but incredibly massive object at the center of our Milky Way galaxy, and suspect it is a black hole. A cloud of gas orbits this object at m/s every 78,500 years.
a) What is the mass of this alleged black hole?
b) How large is this black hole?
A
(a) kg
(b) m
B
(a) kg
(b) m
C
(a) kg
(b) m
D
(a) kg
(b) m
3
example
Replacing the sun with a black hole
Video duration:
5m
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Video transcript
All right guys, let's work this one out together. So the sun is instantly replaced by a black hole, the size of the earth. We're supposed to figure out what the net acceleration of objects on earth's surface is because of this black hole. Now, what exactly does that mean? Well, just let me go, let me go ahead and just draw a little quick little sketch here. Let's imagine. We're standing on the earth's surface and any object on the earth's surface, whether it is it a human being or a little box or whatever, we all are glued to the earth's surface because there is an acceleration downwards that's G earth. And we just know that as 9.8 m per second squared, the idea is that the sun is gonna be replaced by a black hole like this. And due to the mass of that black hole, MBH, it's gonna cause some acceleration upwards. That's gonna be GB H right? So it's gonna cause that acceleration. And so these two accelerations are gonna be fighting each other. And we need to figure out what is the net acceleration of objects on the earth now? Because these two things point in opposite directions. I'm just gonna choose one direction to be positive and negative. So I'm just gonna say that this direction is gonna be positive, which means that the net acceleration is going to be the gravitational acceleration due to the black hole minus the gravitational acceleration due to the earth. That's basically what's happening here, right? Cool. So we actually know what this number is. So what happens is if this number is bigger, then things are gonna get lifted off of the surface, right? So let's go ahead and figure out what this GB H is actually equal to. Let's go ahead and look at that equation. So let's look at our equations for acceleration due to gravity. We have two versions of it when we're on the surface or when we're at a distance of something. Now what happens is we're not standing on the surface of the black hole, we're standing on the surface of the earth. So that means we need to use this equation which is GB H is equal to big G times the mass of the black hole divided by the distance squared, this our distance right here. Now let's go ahead and look, look through our variables. G is just a constant, the mass of the black hole I don't have, but I know what the distance is going to be. This distance here is just the distance between the sun and the earth. So that's RSE and I actually know what that is. That's just this constant over here. So let me just go ahead and highlight that, that's 1.5 times 10 of the 11th. So actually know what this is. Here's the problem. I don't know what the mass of the black hole is. So I'm gonna need another equation to solve this. Let's go ahead and go over here the mass of this black hole. How do we figure that out? Well, we're told the only other information that we're told about this problem about the black hole is that, is that it's earth sized. So we can use the Schwarzschild radius equation to actually relate the size of the black hole with its mass, right? So we have that the Schwarzschild radius equation is just equal to uh we've got two GMB H divided by C squared. So if I'm solving for MBH to plug it back into this equation and then plug this back into this equation, all I have to do is just rearrange. So I've got RS through short star radius times C oops that's not gonna be squared. Sorry, RS times C squared divided by two G. Uh And that's gonna equal the mass of the black hole. So if I could just go ahead and plug everything in there, uh I've got the, the Schwarzschild radius is gonna be the size of the earth. Now, what's the size of the earth? Uh We just have that as 6.37 times 10 to the six. So we're gonna plug that in 6.37 times 10 to the sixth. Now, we've got the speed of light squared that's three times 10 to the eighth. And it's gonna be squared divided by two times the gravitational constant 6.67 times 10 to the minus 11. Now, if you go ahead and plug this in the mass of the black hole is actually going to be uh I get 4.30 times 10 to the 33 kg. So now we're gonna plug this back in and chain it, right. So we've got GB H is equal to, that's gonna be the big constant which, which you know, which we already know what that is times the mass of the black hole, which is gonna be 4.30 times 10 to the 33. We just found that and now we have to divide it by the distance squared, that distance is just 1.5 times 10 to the minus 11. So 1.5 times 10 to the minus 11 or sorry, 1.5 times 10 of the positive of 11. And we have to square that. Now that means that the acceleration due to gravity uh from the black hole is actually going to be 12 point six, you've got 12.6 m per second squared. So now if we plug that back into this equation, the net gravitational acceleration is just 12.6 minus two point or what minus 9.8. And that means that the net gravitational acceleration is going to be 2.8 m per second squared. Notice how this is a positive number. So this is actually our final answer over here 2.8 m per second squared. And because this is a positive number, it means that things are actually gonna get lifted off of the surface of the earth. So that means that yes objects would accelerate off of the earth's surface due to this black hole. All right, let me know if you guys have any questions with this.