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# More astrophysics questions!

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1) If the earth shrinks into a black hole, what will be the radius of the earth?

Solution:

Since the minimum escape speed is v(esp)=c=3x10^8m/s

v(esp)=square root of (2GM/r)

3x10^8=square root of [2(6.67x10^-11)(5.98x10^24)/r)]

r can be found.

As we know, light CAN'T escape from a black hole. But, if we substitute the escape speed as the speed of light, that means speed can escape the black hole's gravitational field. Why can't we substitute c as the escape speed, as the solution does?

2) What is the total amount of energy needed to place a 2000kg satellite in circualr Earth orbit, at an altutide of 500km.

[is the answer simply Et=1/2 Eg=-5.80x10^10 J or is it the change in total energy or something? I don't get what the question is asking for.]

3) The mass of the Moon is 6.7x10^22 kg, and its radius is 1.6x10^6 m. If a woman can raise her centre of gravity 2.0m vertically in a high jump at earth's sruface. How high can she jump with the same muscular effort on the Moon's surface?

[My question is, will the initial velocity of the woman be the same on the earth and on the moon? What I am thinking is that the earth and the moon have different values of Fg, so if she applies the same muscular strength (or same applied force), wouldn't the initial speed as she leaves the ground be different?]

Thanks everyone for explaining! :confused:

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hey kingwinner :shrug:

not to sure what you want to know about the first question.. but number 2 wants to know the change in gravitational potential energy, that will equal the total energy needed to move it to that altitude.

The trick in this one is that at 500km above the surface the gravity due to the earth has dropped to about 8.5, so you cant do a straight E=mgh calculation.

What you really have to do is to intergrate an mg graph against altitude between the two points (earth surface and then earth surface +500km)

3) this is an energy problem, velocity will be different, try calculating how much energy she would need to raise herself 2m of earth and then see how high that will get her on the moon :)

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Hi Kingwinner,

1) I've never heard such a stupid question. Einstein would be turning in his grave. A black hole made out of Earth???? We might as well say, if we were all wierd space creatures from another universe, would we still like chocolate?

However, if you can't find the money to buy a gun to shoot who ever wrote that, they you should consider this. The fastest possible speed anything can go (including light) is C (although anything with mass will never quite reach that speed. This is relativity and it requires at least a first year undergraduate knowledge of physics. Technically, black holes are all about general relativity, but thankfully, not much is asked on that one.

If you really want to know why light will move away from the BH at c, it's because light ALWAYS moves at C. If it misses the black hole, even by a couple of tenths of a milimeter, it will still move at c relative to it. It's all or nothing.

2) Do you remember when I said that a reference point other than at infinity was scientifically meaningless but a good question could nevertheless ask you for the potential energy difference between two points? Well the reason for asking about the DIFFERENCE is that the energy difference is the energy required to get the object from one place to the other.

Jay_qu has given you one correct method to do the calculation, but there is an easier way. 1) work out the GPE of the satalite at the Earth's surface (bog standard question identical to B4). 2) Work out the GPE of the satalite in orbit (same bog standard question). Subtract 2 from 1 (bog standard maths). You should get a +ve answer which means work needs to be done on the satalite.

3) Jay_qu is right. Nothing more to add except that I think the INITIAL velocity will be almost the same. However, it will be slightly more on the moon because the legs will need to overcome less gravity allowing more force on the jump. But this is too much detail for this question. Nevertheless the legs can put in more energy into a jump on the moon than on the Earth, but only slightly. This is because two legs can apply a force of about, say, 7 times your body weight. Gravity reduces your body weight by about a half, say. Therefore, on the moon, the jumping force is about 7.5 times your body weight. In other words, less energy is used in actual the jump on the moon than on the Earth. If you like getting questions wrong by over complicating them, they you should seriously consider these issues.

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Hi Kingwinner,

1) I've never heard such a stupid question. Einstein would be turning in his grave.

To coin a phrase; "There are no such things as stupid questions, only stupid answers". Actually sebbysteiny, given enough pressure a mass the size of the earth could be converted into a black hole. The formula; radius of a black hole = 2G*Mass/c^2 is correct. Remember the theory that suggests there may be mini black holes lurking around that were created shortly after the big bang? Not really a stupid question at all my friend. Questions that cause one to think new and unusual thoughts are anything but stupid, they are the source of most inspired scientific research....................Infy

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1) If the earth shrinks into a black hole, what will be the radius of the earth?

Solution:

Since the minimum escape speed is v(esp)=c=3x10^8m/s

v(esp)=square root of (2GM/r)

3x10^8=square root of [2(6.67x10^-11)(5.98x10^24)/r)]

r can be found.

As we know, light CAN'T escape from a black hole. But, if we substitute the escape speed as the speed of light, that means speed can escape the black hole's gravitational field. Why can't we substitute c as the escape speed, as the solution does?

The solution is correct, but the question is awfully misleading. It is only through a remarkable coincidence of mathematics that the classical and GR escape velocities just happen to be the same. This means that plugging in c as the escape speed will work. Just keep in mind that, in general, physics beyond Newton's laws must apply to black holes.

-Will

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Hi Kingwinner,

1) I've never heard such a stupid question. Einstein would be turning in his grave. A black hole made out of Earth???? We might as well say, if we were all wierd space creatures from another universe, would we still like chocolate?

However, if you can't find the money to buy a gun to shoot who ever wrote that, they you should consider this.

This kind of talk is not accepted here, it will only earn you red boxes as you can probably see. Kingwinner was just asking a question it is unfair for you to speak this way to him, especially when you yourself are incorrect.

Jay-qu

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Hi everyone,

1) A black hole has the property that the escape speed is GREATER than the speed of light. (i.e. light can't escape a black hole)

The solution did it this way, by substituting c as escape speed:

3x10^8=square root of [2(6.67x10^-11)(5.98x10^24)/r)

This means that with a speed of 3x10^8m/s, light CAN escape a black hole, which is not true, right? So is the solution wrong?

2) I was told that Total energy = Eg + Ek, and I can find the total energy at the point where the satellite is in orbit (using the altitude and speed of the satellite). The questions asks for total energy, and I am not sure if this "Total energy" is the required answer...

3) Is it valid to say that the initial velocity will be the same on the earth and the moon and use the kinetic equations to solve this problem?

But the weight is different, the applied force is the same, the time interval of upward acceleration (i.e. to jump) is the same, so wouldn't that initial velocity as she leaves the ground be completely different?

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2. The question asks for the total energy to put it in orbit, and it will be equal to its gravitational plus the kinetic required to give it a circular orbit. Im not sure if you should assume that its initial velocity (due to earth spinning) is negligible..

3. Dont worry about initial velocities. If she could raise herself 2m then she had to put in energy equal to mgh, on the moon g is different, so you end up with:

[math]mg_eh_1 = mg_mh_2[/math] :the mass cancels so

[math]g_eh_1 = g_mh_2[/math]

hence [math]h_2 = (g_eh_1)/g_m[/math]

where [math]g_e = 9.8[/math]

[math] g_m = M_mG/r^2 [/math]

[math] h_1 = 2[/math]

and [math] h_2 = [/math]height jumped on moon

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You make some good points about perfectly good theories of little black holes. I've always been a little skeptical of them, but if they do exist, I doubt they will make much difference to us as they are too small to have a large gravitational effect.

However, an object the size of the Earth can never become a black hole. Maybe an object the size of a proton, or an object the size of a huge star, but nowhere in between.

The Earth simply does not have the gravitational power to push protons and electrons (ie fermions) into high enough energy states for the Earth to collapse into a black hole.

If the there is such a thing as an Earth sized black hole, then you might as well throw all of statistical mechanics and much of quantum mechanics out of the window.

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I've always been a little skeptical of them, but if they do exist, I doubt they will make much difference to us as they are too small to have a large gravitational effect.

If the sun were to become a black hole, it would be about the size of the Earth. Black holes have no definite size limit, but their gravitational influence on spacetime would be exactly the same as other objects of the same mass. Therefore, an Earth-sized black hole would have a huge impact. If it were to pass throuh our solar system it would most likely throw a couple of planets out of their orbits.

If the there is such a thing as an Earth sized black hole, then you might as well throw all of statistical mechanics and much of quantum mechanics out of the window.

What are you basing these claims on?

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I've just read that some people believe my scathing attack on Kingwinner's question was in some way personally offensive.

Whilst I agree that some people will take my comments offensively, I doubt it would be Kingwinner, because it isn't his question.

It seems to me like Kingwinner is just telling us a question he came accross in the A-level course or something similar. It's not his fault that his text book authors or his teacher / examiner thought of such a stupid question. As I said, if it could happen with an Earth sized object it, it would blow statistical mechanics and quantum mechanics out of the window. Although there is no science to prove that little black holes cannot exist (cos it's almost impossible to varify experimentally making such theories REALLY good [note a little sarcasm), there is enough science to know how black holes exist in stars, and therefore that it would break current understanding of the Physics for that to be possible.

Kingwinner seems to me to make a valuable contribution to this forum.

Q1) I think I have already explained why the light moves away at C. The reasoning is that light ALWAYS travels at C under all circumstances from all possible observers. This is special relativity: a topic that is truely awe inspiring.

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I think you are missing something here. It is perfectly acceptable for something with the mass of the earth to become a black hole - it just has to be compacted down, sure it wont do it on its own but if compacted small enough it can become a black hole.

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If the sun were to become a black hole, it would be about the size of the Earth. Black holes have no definite size limit. - Tormod

You might be right, but it would contradict physics as we know it, so I'm not that confidant.

If the sun dies, it WILL NOT BECOME A BLACK HOLE. It would be a red dwarf. Bigger stars become neutron stars. And huge stars become black holes.

What can I say: I've done the calculations and seen the results. If there is not enough gravity then a mass simply cannot push fermions close together enough.

Okay, I'll try and quantify this.

Fermions cannot be in the same place in the same state. Therefore, pushing fermions together can only be done if one fermion moves into a higher energy state. This requires energy, which I will call "Fermi energy" (I can't remember the real name, sorry: don't get this confused with solid state physics). In a star, the gravitational energy balances this "Fermi energy". Further, the force caused by this "Fermi energy" can be calculated because F = dE / dr .

This is the bit my memory is a little hazy on, but this force, at non relativistic speeds, is proportional to r^-1. Thus the size of the star can be found simply by equating the forces of gravity with the force from the "Fermi energy" giving a nice constant value of r.

However, if the star is sufficiently big, then the fermions will be forced into energy levels that consist of relativistic velocities. This changes everything because the force from the "Fermi energy" now becomes proportional to r^-2 or something. This means that r cancels out of the equation and there is no value of r in which the force frome the "Fermi energy" balances the gravitational force. Thus, the star collapses into a ..... neutron star. When protons and electrons get pushed together, they combine to form the heavier neutron. This neutron is also a fermion, and because it is heavier, will be at a non relativistic speed. However, for even bigger stars, even the neutron will be relativistic and the neutron star collapses and with no force capable of preventing collapse, a black hole is formed.

The earth simply is no where near big enough to give protons and electrons relativistic velocities net alone neutrons. Hense the Earth is a planet, not a black hole. Further, no object the size of the Earth can ever be a black hole unless your arguing that the laws of physics as we know them are wrong, but considering the incredible accuracy of statistical mechanics, that will be a tough argument to make.

Black holes have no definite size limit, but their gravitational influence on spacetime would be exactly the same as other objects of the same mass. Therefore, an Earth-sized black hole would have a huge impact.

Your right with the reasoning, but I don't agree with your conclusions. Black holes have no definate size. Having said that, all black holes have an 'event horizen' beyond which the current laws of physics cease to exist, and bigger black holes have bigger event horizens.

Further, the gravitational influence depends on the mass of the object not on the type. Whilst you are right that an Earth sized (in volume) object would have a significant impact on the universe, if the Earth collapsed into a black hole, it would not have any bigger gravitational effect than it has already. It would just be reduced to about the size of a squash ball. Again, we are hypothosising over counterfactual conditions: an object the size of the Earth cannot become a black hole period.

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I think you are missing something here. It is perfectly acceptable for something with the mass of the earth to become a black hole - it just has to be compacted down, sure it wont do it on its own but if compacted small enough it can become a black hole.

Good point. If the Earth can be compacted down, then it might become a black hole. Didn't think of that. However, is it actually possible to compact the Earth that much? The energy required for doing that is absolutely immense. The only way I can think of with that happening is if it is swallowed by a black hole, but then it would not become a black hole, it would join one.

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The solution did it this way, by substituting c as escape speed:

3x10^8=square root of [2(6.67x10^-11)(5.98x10^24)/r)

This means that with a speed of 3x10^8m/s, light CAN escape a black hole, which is not true, right? So is the solution wrong?

No. Just at the point where you compact it to a black hole, the escape speed is exactly c. Here you are on the turning point, but we are looking for the largest radius at which the earth would become a black hole.

Again, something to keep in mind: it is only a coincidence that Newtonian escape velocity formula is the same as the GR escape velocity formula. As such, this application is extremely misleading. You have no reason to suspect your formula should apply to a black hole.

-Will

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2) So basically calculate the change in gravitational potential energy (from ground to altutide of 500km) plus the kinetic energy of the satellite in orbit, am I right? If so, this is basically the work done needed to place a satellite at rest on earth to its orbit, right?

And also, the earth is rotating at a very high speed, which gives the object on the ground a very large kinetic energy. Why can we omit this large amount of kinetic energy in the calculations?

3) Can someone explain how to do question 3? At which 2 points will the total energies be equal? Which 2 points should I use? Thank you!

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