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Why Gravity Gets So Strong Near Dense Objects
相關課程
- In the video on black holes
- several people asked what is actually a pretty good question, which is: If the mass of
- say a black hole is only 2 or 3 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 is 2 or 3 solar masses,
- it's gravity isn't so strong that it keeps light from escaping
- why would a black hole that has the same mass keep light from escaping?
- And to understand that, let's just think a little bit about it.
- I'll just do a Newtonian
- classical physics right here.
- I wont get into the whole general relativity and things.
- This will really just give us the intuition of why a smaller, denser thing with the same mass
- can exert a stronger gravitational pull.
- So let's imagine-- so let's take two examples
- Let's say I have some star here
- that has a mass-- let's just call that mass m1
- and let's say that its radius,
- lets just call this "r"
- and lets say that I have
- some other mass right at the surface
- of this 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?
- You need 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 on the surface
- Now, with that said, let's take another example.
- Let's say this large massive star or whatever it might be, eventually condenses
- into something a thousand times smaller.
- So let me draw it like this.Obviosly I'm not drawing it to scale.
- So let's say we have another case like this (I'm not drawing it to scale)
- So this object, maybe it's the same object or maybe it's a different object,
- but it has the exactly same mass as larger object, but now has a much smaller radius.
- So that radius now is 1/1000.Let's say it's 1/1000 of this radius over here.
- So it's r/1000.If this had a million kilometer radius,
- so that will make it roughly about twice the radius of the sun.
- And this was a million kilometer radius right over here.
- This would be a thousand kilometer radius,
- so maybe we are talking about something that is approaching a neutron star.
- But we don't have to think about what it actually is,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 the same mass that's on the surface of this thing.
- So, this is m2 right over here.
- So, what's going to be the force between these 2 masses?
- What's the force pulling them together?So that's the Universal Law of Gravitation again.
- The force, let's just call this F1(force 1) and let's just call this F2(force 2).
- Once again, it's going to be the gravitation constant, times the product of the 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 are considering m2 to the kind of view
- just to point match right over there.
- So what's the radius squared?
- It's going to be "r" over 1 thousand squared ( r/(1000^2)).
- Or, if we simplify this,what would this be?
- This is the same thing as the gravitational constant times m1 times m2
- over "r" squared over 1 thousand squared or over 1 million.That's just a thousand squared.
- Or, we can multiply the numerator and denominator by 1 million
- and this is going to be equal to 1 million times the gravitational constant
- times m1 times m2, all of that over "r" squared.
- Now, what's this things right over here?
- That's the same thing as the 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 of mass
- So they are both being attracted.They are both exerting this on each other.
- And the reality is that 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.That you know, this just come straight
- from the universal law of gravitation.But what if something closer to this center of mass
- experience the same thing?
- If this is a star, wouldn't photons that are over here wouldn't this experience the same force
- If this distance right here is "r" over a thousand(r/1000), wouldn't some photon here or
- atom here or molecul or whatever it's over here, wouldn't that experience the same force?
- This milllion times the force of this thing and you got to remember,
- all of the 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 is pulling it in outward direction.
- All that mass is doing is that mass itself is being pulled inword.
- 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 me mitigated
- by the fact that there are so much mass over here pulling in the other direction.
- So, you can imagine that if you are in the center of a really massive object,
- there will be no gravitational force being pulled on you,because you are at the center of the mass
- The rest of the mass is outward,so at every point it will be pulling you outward.
- So, that's why you as you enter the core of the star,
- you will get a lot closer to the center of the mass.
- It's not goint to be pulling on you with this type of force and the only way you can get this
- type 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.
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