Alright, 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 ahead and draw a 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's 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 meters per second squared. The idea is that the sun is going to be replaced by a black hole like this. And due to the mass of that black hole, mbh, it's going to cause some acceleration upwards. That's going to be gbh. Right? So it's going to cause that acceleration. And so these two accelerations are going to 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 going to choose one direction to be positive and negative. So I'm just going to say that this direction is going to 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 going to get lifted off of the surface. Right? So let's go ahead and figure out what this gbh is actually equal to. Let's look at our equations for acceleration due to gravity. We have 2 versions of it, when we're on the surface or when we're at a distance of something. 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. That means we need to use this equation, which is gbh is equal to G * (the mass of the black hole) / (the distance squared), this r distance right here. Now, let's go ahead and 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 * 10^11. So, I actually know what this is. Here's the problem. I don't know what the mass of the black hole is. So I'm going to 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 and about the black hole 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 2 * G * mbh / c^2. 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 * c^2 / (2G), and that's going to equal the mass of the black hole. So if I just go ahead and plug everything in there, I've got the Schwarzschild radius is going to be the size of the Earth. Now what's the size of the Earth? We just have that as 6.37 * 10^6. So we're going to plug that in. 6.37 * 10^6. Now we've got the speed of light squared, that's 3 * 10^8, and it's going to be squared, divided by 2 * the gravitational constant, 6.67 * 10^-11. Now, if you go ahead and plug this in, the mass of the black hole is actually going to be, I get 4.30 * 10^33 kilograms. So now we're going to plug this back in and chain it. Right? So we've got gbh is equal to that's going to be the big constant, which we already know what that is, times the mass of the black hole, which is going to be 4.30 * 10^33. We just found that. And now we have to divide it by the distance squared. That distance is just 1.5 * 10^11, and we have to square that. Now, that means that the acceleration due to gravity from the black hole is actually going to be 12.6. You've got 12.6 meters per second squared. So now if we plug that back into this equation, the net gravitational acceleration is just 12.6 minus 9.8. And that means that the net gravitational acceleration is going to be 2.8 meters per second squared. Notice how this is a positive number. So this is actually our final answer over here, 2.8 meters per second squared. And because this is a positive number, it means that things are actually going to 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. Alright? Let me know if you guys have any questions with this.