Dear Long Live Yorke

[LongLiveYorke 38]

http://whistleralley.com/planimeter/planimeter.htmHelpI need this thing called a "planimeter" to figure out approximate land areas on plat maps of old county real property records, but I don't understand how it works as it involves symbols and something called "Green's theorem".

Well, I'm not sure that you would practically need to know the mathematics behind how it calculates areas to actually use it, but I'm glad that you're curious for the sake of curiosity.It "sort of" involves Green's theorem. Green's theorem is one of the most important results in Calculus (and is a specific case of Stoke's theorem). In general, it states that if we ahve some shape, then we can relate a function on the boundary of the shape to the function throughout the region of the shape. This is an extremely crude description, but to be more exact, we'd need calculus (and maybe some Topology and Manifolds to make it really fun). I'm not exactly sure that Green's theorem is used in this situation. I can see how someone would think it, since it is very similar to situations that use Green's theroem. However, Green's theorem requires knowledge of the tangent of the curve around the boundary, and this is not exactly what we're given. I think the website you linked gives a pretty decent description, though it's certainly not fantastic and the author really isn't great at explaining things. The man idea is that we can determine any point in the plane by two numbers. There are many pairs of numbers that we can give to convey the same information. Usually it's easiest to give x-y information (ie north/east, up-down/left-right, etc etc). But we can also give many other numbers. Using this device, we see that we can give the position of a point using two numbers: the two angles that are made by the two arms of the machine. Fine, not so difficult to understand. If you give me exactly how to bend the two arms of the machine, you will determine a unique point in its range on the map or whatever. So, if we record the angles of every point along the boundary, we will fully know the shape of the boundary and therefore (in theory using Green's theorem as proof) could determine the area.However, this device doesn't record the two angles. Rather, it only records the motion of one of the arms. So, the question is, "Can we get enough information about the perimeter to determine the area using only this wheel that gives us a certain motion of one of the arms?" The answer, of course, is yes (of course only because this device exists and people actually use it). The wheel determines the angle moved by the outer arm. Assuming it doesn't slip, the rolling of the wheel is in one to one coorespondence with the angle traversed by the second (outer) arm. The whole point is that the area in general must be expressed as an integral which will depend on the two angle variables. The article demonstrates that this integral seperates into the sum of two integrals, each of only one of the variables. Further, the inner angle integrates to zero since if you end up back at the same point, the total distance that the inner angle has traversed is zero. I don't think it's particularly obvious that the area integral easily seperates into the sum of two integrals, and you need a little geometry and calculus to show this. I'm not sure if this answered your question at all. Let me know if something's uncear and I'll have another go at it. This would be easier if I had a blackboard in front of me.

[KramitDaToad 0]

I have a question that I'm too lazy to find an answer for ;)As I understand it, in nucleosynthesis the energy comes from mass being converted to energy , however if you take any of the reactions that end in He-4 for example you have the same amount of atomic particles going in as coming out. Where does the extra mass come from?I suppose another way to ask would be - Why is He-4 not twice as heavy as deuterium?

[LongLiveYorke 38]

I have a question that I'm too lazy to find an answer for ;)As I understand it, in nucleosynthesis the energy comes from mass being converted to energy , however if you take any of the reactions that end in He-4 for example you have the same amount of atomic particles going in as coming out. Where does the extra mass come from?I suppose another way to ask would be - Why is He-4 not twice as heavy as deuterium?

The differences in mass come from binding energies. Helium-4 has exactly twice the amount of particles that deuterium has (two protons, two neutrons, two electrons). However, in order to have two protons in the nucleus, they must be held together by a strong binding force since electromagnetically they would want to repel each other. So, the strong force holds them together and does so with a lot of energy. The strong force holds the constituent particles of the nucleus together with a considerable amount of (negative) energy (it is negative because we would have to do work on the nucleons to pull them apart). So, if we want to think in terms of masses, we can say that this negative binding energy adds to the effective mass through m = E/c^2But I think it's often confusing to think in terms of mass. It's better to ignore the concept of mass and think in terms of rest energies. Thus, the rest energy of the nucleus is clearly less than the sum of the indivudual rest energies of its constituent particles because there needs to be a strong negative potential to hold the protons and neutrons together.

[rdtedm 0]

I love this thread. Then again, I'm an applied math major :/

[LongLiveYorke 38]

I love this thread. Then again, I'm an applied math major :/

Wait, you can apply math to real things? Amazing!

[KramitDaToad 0]

The differences in mass come from binding energies. Helium-4 has exactly twice the amount of particles that deuterium has (two protons, two neutrons, two electrons). However, in order to have two protons in the nucleus, they must be held together by a strong binding force since electromagnetically they would want to repel each other. So, the strong force holds them together and does so with a lot of energy. The strong force holds the constituent particles of the nucleus together with a considerable amount of (negative) energy (it is negative because we would have to do work on the nucleons to pull them apart). So, if we want to think in terms of masses, we can say that this negative binding energy adds to the effective mass through m = E/c^2But I think it's often confusing to think in terms of mass. It's better to ignore the concept of mass and think in terms of rest energies. Thus, the rest energy of the nucleus is clearly less than the sum of the indivudual rest energies of its constituent particles because there needs to be a strong negative potential to hold the protons and neutrons together.

Cooool. Thanks for thatThe thing I fail to comprehend though is why the strong force is a negative energy (not sure if I worded that well). I'm not even sure how you get negative energy. I imagine it as 2 balls that repulse each other being pushed together. You need some force to do that, hence the high temperatures and pressure to kick start, but even once they are together sure they are still electromagnetically repulsing each other and you would need a constant effort to overcome that. In my simple view that would make the situation a consumer of energy. What am I missing?

[LongLiveYorke 38]

Cooool. Thanks for thatThe thing I fail to comprehend though is why the strong force is a negative energy (not sure if I worded that well). I'm not even sure how you get negative energy. I imagine it as 2 balls that repulse each other being pushed together. You need some force to do that, hence the high temperatures and pressure to kick start, but even once they are together sure they are still electromagnetically repulsing each other and you would need a constant effort to overcome that. In my simple view that would make the situation a consumer of energy. What am I missing?

So, I thought about it a little bit, and my answer from before really wasn't satisfactory. It is the standard textbook answer, but it really doesn't make all that much sense, so I'll try to be a bit clearer.Negative energy isn't anything special and certainly isn't specific to the strong force. Mainly, it's convention and it's a convention that could potentially be confusing. So let's see if we can make sense of it.Imagine that I have two asteroids separated at opposite ends of the universe. What are the energies of these asteroids? To be complete, we'll follow Einstein and include all forms of energies. The energy is the rest energy plus the kinetic energy plus the potential energy. Now, if you will remember from high school physics, potential energy is an arbitrary quantity, and all that really matters is change in potential energy. It is convention to set potential energies to zero at "infinity," meaning that as we separate two things toward infinity, their potential energies will approach zero. Fine. So, these asteroids have some mass which makes up their rest energy, they are standing still so they have no kinetic energy, and they are separated by infinity so they have no gravitational potential energy. Their energies are just the sum of their masses.Now, imagine that I give one of them a slight kick, just enough to get them moving toward one another but not enough to add considerable energy. So, the asteroids will slowly move toward each other, and as they do so, the force of gravity between them will get stronger and stronger, so they will move faster and faster toward one another. They will pick up speed and gain kinetic energy. This kinetic energy comes from potential energy, so they must be losing potential energy (in other words, potential energy is being converted to kinetic energy). So, the potential energy, which was before zero, must now be negative. But again, this is simply a convention. We could have decided that their initial potential energies was some arbitrarily large number so that it will remain positive throughout this thought experiment. The point is that the potential energy goes down and the kinetic energy goes up.So, what happens. They are going very fast, they approach one another, and now let's imagine that they get very close but don't actually hit each other. So, at one point, they will have maximized their kinetic energy when they're closest to each other, and then they will fire past each other and shoot back into space. Since energy is conserved, they will be able to again go infinitely far past one another and back to opposite ends of the universe since they had enough kinetic energy to escape each other's gravitational potential (and we know this because they started infinitely far away. If they couldn't make it infinitely far away, then somewhere energy was lost).Now, let's try it again. But this time, when they're pretty close to each other, imagine that one of them maybe goes through a dust cloud. It hits the dust and is slowed down. In the process of slowing down the asteroid, the asteroid loses some kinetic energy. This energy must therefore be given to the dust somehow since energy is conserved (maybe it is given in the form of heat and the dust becomes hotter, but you can imagine whatever you'd like). So now one of the asteroids has lost maybe a considerable amount of energy. When they come to each other again, they will no longer be able to escape back to infinity because one of them no longer has sufficient kinetic energy to escape the potential energy. Instead, maybe they now orbit each other. They have become a two asteroid system and they spiral around their common center of mass. Cool.So, the question is, what is the mass of this system? Well, we know that mass and energy are the same thing. Rather, rest mass is a form of energy, and often the biggest contributor to the energy of an object (unless the object is relativistic, and this is really the definition of being relativistic). So, how do we measure mass. Mass is measured in two ways: either by measuring it's coupling to gravity (by putting it on a scale and weighing it, for instance), or by measuring how it resists force (using f=ma, you move it back and forth and see how hard it is to do so). But in relativity, these two mechanisms don't measure the rest mass of an object, they measure its total energy. Now, if an object is at rest completely, they will measure its rest mass, so if you put a ball on a scale, you will measure what we normally consider to be its mass which is its rest mass or rest energy or whatever you'd like to call it. But if the ball is moving fast or if the ball is really hot and therefore has a lot of internal kinetic energy, it will appear heavier since the total energy will be greater.So, what of our system. Will it weigh more than the sum of the rest masses of the two asteroids, less, or exactly the same. Well, if we agree that the mass we can measure is really total energy of the system, then we would expect it to weigh less. Why? Because at a certain point, to make a bound system, we had to remove energy from one of the asteroids (when it went through the dust). So, the total energy of this bound system is less than the energy of the two asteroids sitting far away in space and they will weigh less. The same thing occurs with protons and neutrons. When they bind together to form a nucleus, they give off energy in the process of binding, and therefore the nucleus weighs less than the sum of the two component nucleons.Now, if we agree that the potential energy of the system is zero at infinity and gets more and more negative as they approach being close together, then when we remove the kinetic energy necessary to bind the two asteroids together, we will have there be more potential energy than kinetic energy and the total energy of the system will be negative (not including the rest energy from mass, which is really most likely much more energy than either the kinetic or potential energies). So, this is potentially somewhat confusing but really useful. If the energy of a system is negative (that is, kinetic plus potential is negative), then it is a "bound" system and the constituent particles will not be able to escape one another. So, this is what I meant above. The nucleus is a bound system and so its kinetic plus potential energy is negative, and so it therefore has less energy/mass than if the particles were separated. So, we can think that this negative binding energy contributes to negative mass of the system via E=mc^2, and that's what I said in my above post, but I think it makes more sense to really describe what is going one as I just did and to simply say that it has less energy because, in order to bind it together, at some point energy must be removed.

[speedz99 145]

The same thing occurs with protons and neutrons. When they bind together to form a nucleus, they give off energy in the process of binding, and therefore the nucleus weighs less than the sum of the two component nucleons.

I'm a little lost here. It makes more sense to me that the energy of the bonds between the nucleons would make the nucleus weigh more, not less, than the components. Why do they give off energy in the process of binding, wouldn't it be an endergonic (if that's the right term in this situation) reaction?

[LongLiveYorke 38]

I'm a little lost here. It makes more sense to me that the energy of the bonds between the nucleons would make the nucleus weigh more, not less, than the components. Why do they give off energy in the process of binding, wouldn't it be an endergonic (if that's the right term in this situation) reaction?

This is the idea behind nuclear fusion. You take two nuclei and smash them together to form one nucleus. In the process of binding together to form one nucleus, energy is given off. This means that the single nucleus as a system has less energy than the combined energy of the two other nuclei. So, a lot of energy can be given off. This is because the strong force is very powerful (the strong force is the force that holds nuclei together and binds protons and neutrons together [well, really it binds quarks together, but that's a different story all together]). In term, if we then wanted to separate the two nuclei, we would need to give them each a lot of energy (exactly the same amount that they give off when they fuse). It's potentially confusing to think about the "bonds" having mass. It makes more sense if you always think in terms of energy. If in creating a system, you have to take energy away from it, it will weigh less as a system than as components. If you need to add energy, it will weigh more. Trying to picture the mass stored in a gravitational field or in an electromagnetic field or, in this case, in the quantum chromodynamic field can be confusing. Since energy is conserved, the only way that energy/mass can change is if it is released or lost somewhere. So, in the first case of the meteors when energy wasn't lost to dust, they simply moved past each other and went back into space. If you had measured the mass of the unbound system, it would have been the same as the mass of the component meteors since energy wasn't lost anywhere.

[speedz99 145]

Trying to picture the mass stored in the quantum chromodynamic field can be confusing.

Amen, brother.

[KramitDaToad 0]

<lots of smart stuff>

Fantastic! That makes a lot more sense.Great analogy and thanks for all the effort.

[speedz99 145]

I realize this might belong in the cat thread, but...

[hblask 1]

In school we are taught that spacecraft, upon re-entry, get very hot because of friction with the atmosphere. I also heard a scientist explain once that it's not friction, it's due to the compression (all else being equal, compressed air is warmer than uncompressed air), with the idea being that coming in so fast the air beneath the spaceship can't get out of the way and gets compressed. To me, the air compression theory makes sense, but I've heard arguments on both sides. Which is it? Does anyone even know?

[LongLiveYorke 38]

In school we are taught that spacecraft, upon re-entry, get very hot because of friction with the atmosphere. I also heard a scientist explain once that it's not friction, it's due to the compression (all else being equal, compressed air is warmer than uncompressed air), with the idea being that coming in so fast the air beneath the spaceship can't get out of the way and gets compressed. To me, the air compression theory makes sense, but I've heard arguments on both sides. Which is it? Does anyone even know?

Air really doesn't have that much friction at all (which makes sense, because it's air). So, I'm very willing to believe that it's friction. When air causes "drag," it isn't the friction between the air and the object, it's really the fact that the object stirs the air around, makes little vortices, and it takes energy to make vortices spinning. Designes on airplanes and cars aren't made to reduce friction, they're made to reduce the amount of little air whirlwinds (vortices, eddies, etc) that the object makes. So, I'm thinking that it is indeed the compression of the air which causes its temperature to rise (simple application of the ideal gas law), which in tern heats up the ship.

[myenemy 0]

What are your thoughts on aether? (You know the medium in space by which everything is relative to. I heard it from ABHOT) I think its a bunch of hooey!

[LongLiveYorke 38]

What are your thoughts on aether? (You know the medium in space by which everything is relative to. I heard it from ABHOT) I think its a bunch of hooey!

The aether is a general term that has been applied historically thought physics to describe certain things we don't understand. It most famously was invented to create a medium through which light moved. People used to believe that light was a wave, akin to a water wave, and therefore it had to move through something (so the aether was the water to light's wave). There was an extremely famous experiment called the Mickelson Morely experiment, which famously found a null result when looking for the aether. This experiment was very important for the early days of relativity. I'm not sure exactly to what you're referring to here when you discuss the aether. Einstein compared his notion of "Space" in general relativity to a sort of "aether" through which matter moves. The bottom line is that the term aether has no real specific meaning and most meanings that have been given to it are, as you would say, a bunch of hooey.

[tobytobey 0]

I'm not sure if I feel more/less smarter/dumber after reading this thread.

[Vandees 0]

Dear LLY,Which, in your educated opinion, came first, the chicken or the egg? And as a follow up, how? Thank you.

[speedz99 145]

The evolutionary ancestor of the chicken laid eggs.

[simo_8ball 0]

The evolutionary ancestor of the chicken laid eggs.

God made the chicken, which then laid an egg.BET YOU CAN'T PROVE ME WRONG

[myenemy 0]

Is time travel now, or ever going to be, possible?This is fun: http://en.wikipedia.org/wiki/Ontological_paradox

[LongLiveYorke 38]

Is time travel now, or ever going to be, possible?This is fun: http://en.wikipedia.org/wiki/Ontological_paradox

Short answer:Traveling forward in time- yes (it's quite easy)Traveling backwards in time- no (because then you could kill your own grandfather and so on. These sort of acausal paradoxes are a no-no in most physicist's interpretation of how the universe works).

[LongLiveYorke 38]

By the way, I'm currently spending the summer in CERN working on the ATLAS experiment. Although the weather has been somewhat horrible, the atmosphere is very exciting as things are starting to be turned on and cooled down. I just finished participating in a dress rehearsal for the data acquisition and monitoring that will begin at full scale hopefully toward the end of August and into September. I'll let you know if we create a black hole and destroy the world.

[speedz99 145]

Traveling forward in time- yes (it's quite easy)

Using relativity doesn't count. I assume that's what you mean.edit: I'm referring to using relativity by just jumping on a ship and going really fast. I feel like that's not time travel by the popular definition of the phrase.

Traveling backwards in time- no (because then you could kill your own grandfather and so on. These sort of acausal paradoxes are a no-no in most physicist's interpretation of how the universe works).

I don't accept "because it would create paradoxes" as a reason why it's not possible. Maybe we just don't understand how the paradoxes would get resolved.I watched some show recently about some guy who thinks that he can create a machine that will allow things (starting with particles) to go back in time, but only to a time after the machine was turned on. So if he perfects it (which I'm sure won't happen) and turns it on, a person living 2,000 years from now could travel back. There was also the implication that an alien race could have made such a machine millions of years ago, so if we ever contact them we could go back to any time in our history. I wish I could remember his name...he was an african american and just wrote a book about his life in physics, if that helps (I feel like that can't describe too many people).

[Sal Paradise 57]

I watched some show recently about some guy who thinks that he can create a machine that will allow things (starting with particles) to go back in time, but only to a time after the machine was turned on. So if he perfects it (which I'm sure won't happen) and turns it on, a person living 2,000 years from now could travel back. There was also the implication that an alien race could have made such a machine millions of years ago, so if we ever contact them we could go back to any time in our history. I wish I could remember his name...he was an african american and just wrote a book about his life in physics, if that helps (I feel like that can't describe too many people).

so politically correct, then sooooo not politically correct. nice.