Why gravity gets so strong near dense objects

In the video on black hole several people asked what is actually a pretty good question, which is if the mass of say a black hole is only two or three solar masses, why is the gravity so strong? Obviously the sun's gravity isn't so strong that it keeps light from escaping, so why would something, or even a star that's two or three solar masses-- its gravity isn't so strong that it keeps light from escaping. Why would a black hole that has the same mass, why would that keep light from escaping? And to understand that, I'll just do Newtonian classical physics right here. I won't get into the whole general relativity of things. And this really will just give us the intuition of why a smaller, denser thing of the same mass can exert a stronger gravitational pull. So let's take two examples. Let's say I have some star here that has a mass m1. And let's say that its radius, let's just call this r. And let's say that I have some other mass right at the surface of the star, somehow able to survive those surface temperatures. And this mass over here has a mass of m2. The universal law of gravitation tells us that the force between these two masses is going to be equal to the gravitational constant times the product of the masses. So m1 times m2, all of that over the square of the distance r squared. Now, let me be very clear. You might say, wait, this magenta mass right here is touching this larger mass. Isn't the distance 0? And you have to be very careful. This is the distance between their center of masses. So the center of mass of this large mass over here is r away from this mass that's on the surface. Now, with that said, let's take another example. Let's say that this large massive star, or whatever it might be, eventually condenses into something 1,000 times smaller. So let me draw it like this. And obviously I'm not drawing it to scale. So let's say we have another case like this. And I'm not drawing it to scale. So this object, maybe it's the same object or maybe it's a different object, that has the exact same mass as this larger object, but now it has a much smaller radius. Now the radius is 1/1,000 of this radius over here. So maybe I'll just call it r/1,000. So if this had a million kilometer radius, so that would make it roughly about twice the radius of the sun, if this was a million kilometer radius right over here, this would be 1,000 kilometer radius. So maybe we're talking about something that's approaching a neutron star. But we don't have to think about what it actually is. Let's just think about the thought experiment here. So let's say I have this thing over here. And let's say I have something on the surface of this. So let's say I have that same mass that's on the surface of this thing. So this is m2 right over here. So what is going to be the force between these two masses? How strong are they going to want to-- What's the force pulling them together? So let's just do the universal law of gravitation again. Let's just call this force one, and let's call this force two. Once again, it's going to be the gravitational constant times the product of their masses. So the big m1 times the smaller mass, m2, all of that over this distance squared, this radius squared. Remember, it's the distance to the center of masses. This center of mass here, we're considering m2 to kind of be just a point mass right over there. So what's the radius squared? It's going to be r/1,000 squared. Or if we simplify this what will this be? This is the same thing. And I'll just write it in one color, just because it takes less time. Gravitational constant m1 m2 over r squared over 1,000 squared, or over 1 million. That's just 1,000 squared. Or we can multiply the numerator and the denominator by 1 million, and this is going to be equal to 1 million-- I'm going to write it out, 1 million. Let me scroll to the right a little bit-- times the gravitational constant, times m1 m2, all of that over r squared. Now, what is this thing right over here? That's the same thing as this F1. So this is going to be 1 million times F1. So even though the masses involved are the same, this yellow object right here is the same mass as this larger object over here. It's able to exert a million times the gravitational force on this point mass. And actually vice versa. They're both being attracted. They're both exerting this on each other. And the reality is, because this thing is smaller, because this m1 on the right here, this one I'm coloring in, because this one is smaller and denser, this particle is able to get closer to its center of mass. Now, you might be saying, OK, well, I can buy that. This just comes straight from the universal law of gravitation. But wouldn't something closer to this center of mass experience that same thing? If this was a star, wouldn't photons that are over here, wouldn't this experience the same force? If this distance right here is r/1,000 wouldn't some photon here, or atom here, or molecule, or whatever it's over here, wouldn't that experience the same force, this million times the force as this thing? And you've got to remember, all of a sudden when this thing is inside of this larger mass, what's happening? The entire mass is no longer pulling on it in that direction. It's no longer pulling it in that inward direction. You now have all of this mass over here. Let me think of the best way that's doing it. So you can think of it all of this mass over here is pulling it in an outward direction. It's not telling. What that mass out there is doing, since that mass itself is being pulled inward, it is pushing down on this. It is exerting pressure on that point. But the actual gravitational force that that point is experiencing is actually going to be less. It's actually going to be mitigated by the fact that there's so much mass over here pulling in the other direction. And so you could imagine if you were in the center of a really massive object-- so that's a really massive object. If you were in the center, there would be no net gravitational force being pulled on you, because you're at its center of mass. The rest of the mass is outward. So at every point it will be pulling you outward. And so that's why if you were to enter the core of a star, if you were to get a lot closer to its center of mass, it's not going to be pulling on you with this type of force. And the only way you can get these types of forces is if the entire mass is contained in a very dense region, in a very small region. And that's why a black hole is able to exert such strong gravity that not even light can escape. Hopefully that clarifies things a little bit.