If you place any two masses near each other, their mutual gravitational attraction draws them together. That is, they attract each other. Let us simplify the problem further, and assume one of the objects is of mass M, and the other is much, much less massive. They both still attract each othert, but we have picked a situation where the more massive attracts the other one, and the attraction of the smaller for the larger is so insignificant as to be unmeasurable. For example, a 100kg artificial satellite orbiting earth. The resulting equation is
g = GM/r**2
where g is the gravitational acceleration of the small mass towards the earth,
r is the distance from the satellite to the center of the earth,
M is the mass of the earth
and G is a constant ( a number used to make everything come out in the right units).
So for example, if you drop a brick off a building, M is the mass of the earth in kilograms, r is the distance from the brick to the center of the earth, in meters, and G is the Gravitational Constant for metric units. The gravitational acceleration g experienced by the brick is 9.8 meters/sec/sec or 9.8 m/s**2. This means the first second after being dropped the brick will reach a speed of 9.8 m/s. After 2 seconds, it will achieve a speed of 19.6 m/s, and so on.
Why is r measured to the center of the earth? After all, the building may just be a few meters tall, and the matter in the ground attracting the brick is pretty close to it. The matter on the other side of the earth is also attracting the brick, but it is much further away. In fact, every particle of matter on the whole planet is attracting that brick by different amounts depending on how far away it is, and how far off center it is located to the brick.
As it turns out (and this is the most important factoid in this little speech!) The combined attractions of every atom of the earth attracting that brick act as if they were all concentrated at a point located at the very center of the earth. You can prove this with calculus, but take my word for it: its true. As the equation suggests, all you need to know to calculate the acceleration on the brick is the total mass of the earth, and how far away the brick is from the earth’s dead center.
Since there is not much difference in r between the top and bottom of the building, g is about the same at the top of the building as it is at ground level. But if you could build a tower 4000 miles high (one earth radius) you would now be twice as far away from the center of the planet, and the gravitational acceleration would be 1/4 what it is at the surface. in our equation, r is doubled, so 1/r**2 is one fourth as big. That square exponent changes things in a hurry.
So the gravitational acceleration varies as the mass of the earth, and as the inverse square of the distance to the center of mass of the planet. So as the brick falls off our big tower, not only does it speed up, but its acceleration increases as well. The equation gives you the instantaneous acceleration at every point. To calculate its speed at that point you need calculus.
Now you would think that if you could magically transport yourself to a cave at the very center of the earth, r would be 0 and your acceleration would be infinite. But it doesn’t work that way. Under the earth’s surface, some atoms are pulling you towards the center, but they are counteracted by atoms between you and the surface. At the centroid, you are being simultaneously accelerated in all directions, they would all cancel out, so you would be weightless.
So what’s this got to do with black holes? Well, lets imagine our sun (its mass is, of course, 1 solar mass, by definition) were to suddenly collapse into a black hole just a few miles across. Would the earth be sucked into the hole, along with the rest of the solar system? No, because our distance to the sun’s center of mass is still 1 astronomical unit. The newly formed black hole would still mass one solar mass. The gravitational acceleration on earth due to the sun would remain exactly the same. The earth and the rest of the solar system would continue orbiting the new dark sun exactly as they always did. The great increase in gravitational acceleration associated with black holes isn’t because they are so much heavier, its because they are so small you can get a lot closer to their centers.
Lets go back to the earth, and our building and brick example. Suppose you could magically squeeze the earth to half is current size, ( 2000 miles radius, not 4000) without altering its mass.
If you now stood on its surface, the mass of the earth M is still the same, but your distance r is now half as much. Your gravitational acceleration g is now 4 times as much: 39.2 m/s**2. Since “weight” is mass times acceleration, a 200 pound man would now weigh 800 pounds! You can see how the gravity at the surface of a small body can quickly rise to crushing strength as it shrinks, even if it remains the same mass.
So to sum up, don’t think of black holes as sucking up everything in sight. Any star that becomes a black hole still has the exact same gravitational attraction to distant objects as it did before. Only when you approach closer than what used to be the surface radius of the original star does the acceleration start climbing out of sight.