Black Holes - Online Tutor, Practice Problems & Exam Prep

Black holes are incredibly dense objects with such strong gravitational pull that not even light can escape. The Schwarzschild radius defines the size of a black hole, calculated using the equation $2^G\frac{M}{c^2}$. Anything crossing the event horizon is trapped forever. For example, a black hole with a Schwarzschild radius of 120 AU has a mass approximately 6 billion times that of the Sun, illustrating the immense density and gravitational influence of black holes in the universe.

Black Holes

Hey, everyone. So in this video, we're going to talk about some fascinating objects in our universe which are called black holes. Alright? 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, alright? 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, it's an incredibly small distance, alright? 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 star or something like that or a spaceship or even light itself, once it falls in, it can never come back out. And the reason for that is if you've seen escape velocity, it's okay if you haven't. Basically, the escape speed would have to be faster than the speed of light, which is 3×108 m/s, but nothing can go faster than light. So essentially, anything that falls in is just doomed and will never be able to come back out. And that's why it looks black to us because no light escapes it and actually reaches our eyes or our telescopes and stuff like that. Alright? 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 from 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 2GM_BH/c2. 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, you would feel like a barrier or something like that. But basically, what happens is that once you pass across this boundary, you would 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. Alright? So everything else that we've 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 Schwarzschild radius equation. Alright? 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 MEDIT statement. By the way, this is a real thing that happened. And what they determined is anything closer than this distance here, a 120 AU, falls in and never escapes. Now, you should recognize that as basically the distance in which something comes into the black hole and then never comes back out. So this is going to be the Schwarzschild radius. We want to calculate the mass of this black hole, but we want it in terms of solar masses, in terms of basically as a multiple of how massive our sun is. Okay, so we want the M_BH, but we want it in terms of M_sun, right? So if we want the mass of the black hole, we're going to have to use our new equation, the Schwarzschild radius. So this is R_S = 2GM_BHc2. If we want this M_BH, we're just going to have to rearrange. The c2 goes up to the top. The 2Gs come down, and you're going to have R_Sc2 × c2 / 2 G, and this is going to equal the mass of your black hole. Alright, so we know what the Schwarzschild radius is, it's the 120 AU, but because we're calculating the mass, we need everything to be in SI units. So this R_S here, which is 120 AU, we're going to 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 going to have to cancel out AU on the bottom. So this is going to be 1.5×1011 meters, and that's going to get rid of your AUs. Alright, so this just becomes 1.8×1013 meters. Okay? So now this is going to be 1.8×1013 and then you're going to do 3×108, that's the speed of light but you have to square it and now you divide by 2 × 6.67 × 10 minus 11 that's your big G, and then when you work this out, what you're going to get here is you're going to get a number that's 1.21×1040 kilograms. Alright? Now, this might not seem so, you know, it might seem like any ordinary big number. But remember, we want to express this in terms of solar masses. Okay. So there's one last conversion I need to do, which is I want to convert these kilograms here to M_sun. Alright, so what I have to do is I'm going to have to multiply this so that the kilograms cancel on the bottom. So this is going to be 2×1030. This is going to be 1. This is going to be M_sun. So when you work this out, your kilograms should cancel and the mass of your black hole is going to be 6×109 M_sun. So if you, so for those of you who, realize what this number is, this is going to be about 6,000,000,000. So 6,000,000,000 times the mass of our sun. So, in other words, this black hole is 6,000,000,000 times heavier than our Sun.

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 $1.8\times 10^5$ 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) $3.45\times 10^{37}$ kg (b) $5.11\times 10^{10}$ m
• B. (a) $3.45\times 10^{37}$ kg (b) $7.1\times 10^{16}$ m
• C. (a) $1.92\times 10^{34}$ kg (b) $7.1\times 10^{16}$ m
• D. (a) $1.92\times 10^{34}$ kg (b) $5.11\times 10^{10}$ m

Replacing the sun with a black hole

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, m_bh, it's going to cause some acceleration upwards. That's going to be g_bh. 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 g_bh 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 g_bh 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 r_se. 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 * m_bh / c^2. So if I'm solving for m_bh 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 r_s * 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 g_bh 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.

Here’s what students ask on this topic:

What is a black hole and how is it formed?

A black hole is an astronomical object with an extremely strong gravitational pull, so intense that not even light can escape from it. Black holes are formed when massive stars exhaust their nuclear fuel and undergo gravitational collapse. This collapse compresses the star's core to a point of infinite density known as a singularity, surrounded by an event horizon, the boundary beyond which nothing can escape. The size of a black hole is defined by its Schwarzschild radius, which depends on its mass.

What is the Schwarzschild radius and how is it calculated?

The Schwarzschild radius is the radius of the event horizon of a black hole, beyond which nothing can escape its gravitational pull. It is calculated using the equation:

$R_S=\frac{2GM}{c^2}$

where $G$ is the gravitational constant, $M$ is the mass of the black hole, and $c$ is the speed of light.

What happens to objects that fall into a black hole?

Objects that fall into a black hole cross the event horizon and are trapped forever. The gravitational pull is so strong that they cannot escape, even if they travel at the speed of light. As they approach the event horizon, they experience extreme tidal forces that can stretch and compress them, a process known as spaghettification. Once inside the event horizon, they move inexorably towards the singularity, where they are crushed to infinite density.

How do astronomers detect black holes if they emit no light?

Astronomers detect black holes by observing their effects on nearby objects and light. For example, they can detect the gravitational influence of a black hole on the orbits of nearby stars or gas clouds. Additionally, when matter falls into a black hole, it heats up and emits X-rays, which can be detected by telescopes. The presence of an event horizon can also be inferred from the absence of light from the region around the black hole.

What is the significance of the event horizon in a black hole?

The event horizon is the boundary around a black hole beyond which nothing can escape. It is significant because it marks the point of no return; once an object crosses this boundary, it is inevitably pulled into the black hole. The event horizon is also where the escape velocity equals the speed of light, making it impossible for anything, including light, to escape. This is why black holes appear black, as no light can reach an observer from within the event horizon.