Holy Family Visitation Public School Kattoor.
Science & Maths Club Inauguration on 23 June 2014
Inaugurated by Prof. K.A Solaman
Seminar Conducted by Dr. Sibi K.S
Thursday, June 26, 2014
Science & Maths Club Inauguration-Holy Family Visitation Public School Kattoor.
Holy Family Visitation Public School Kattoor.
Science & Maths Club Inauguration on 23 June 2014
Inaugurated by Prof. K.A Solaman
Seminar Conducted by Dr. Sibi K.S
Tuesday, June 24, 2014
അത് ദൈവകണം !
അത് 'ദൈവകണം' തന്നെ
2012 ല് സേണിലെ കണികാപരീക്ഷണത്തില് കണ്ടെത്തിയത് ഹിഗ്ഗ്സ് ബോസോണ് തന്നെയെന്നതിന് കൂടുതല് സ്ഥിരീകരണം
http://goo.gl/LJoSqB
http://goo.gl/LJoSqB
Monday, June 23, 2014
Sunday, June 22, 2014
Friday, June 20, 2014
Monday, June 16, 2014
The Lorentz transformations
In physics, the Lorentz transformation (or transformations) is named after the Dutch physicist Hendrik Lorentz. It was the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before special relativity. The transformations describe how measurements of space and time by two observers are related. They reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events. They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much smaller than the speed of light. The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the Lorentz transformations preserve the spacetime interval between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
Zero-point energy
What is the 'zero-point energy' (or 'vacuum energy') in quantum physics? Is it really possible that we could harness this energy? SOURCE Scientific American
The Zero Point Energy (ZPE) is an intrinsic and unavoidable part of quantum physics. The ZPE has been studied, both theoretically and experimentally, since the discovery of quantum mechanics in the 1920s and there can be no doubt that the ZPE is a real physical effect. The "vacuum energy" is a specific example of ZPE which has generated considerable doubt and confusion. In a completely empty flat universe, calculations of the vacuum energy yield infinite values of both positive and negative sign--something that obviously does not correspond to the nature of the real world. Observation indicates that in our universe the grand total vacuum energy is extremely small and quite possibly exactly zero. Many theorists suspect that the total vacuum energy is exactly zero. It definitely is possible to manipulate the vacuum energy. Any objects that change the vacuum energy (electrical conductors, dielectrics and gravitational fields, for instance) distort the quantum mechanical vacuum state. These changes in the vacuum energy are often easier to calculate than the total vacuum energy itself. Sometimes we can even measure these changes in the vacuum energy in laboratory experiments. In classical physics, if you have a particle that is acted on by some conservative force, the total energy is E = (1/2) mv2 + V(x). To find the classical ground state, set the velocity to zero to minimize the kinetic energy, (1/2)m v2, and put the particle at the point where it has the lowest potential energy V(x). But this result is only a classical approximation to the real world. Because the classical ground state completely specifies both the particle's speed (zero) and position (at the minimum), it violates the famous Heisenberg Uncertainty Principle (m dv dx > hbar). Quantum physics, via the Uncertainty Principle, forces the particle to spread out both in position and velocity and so causes it to have an energy somewhat higher than the classical minimum. The ZPE is defined as this shift: E(ZPE) = E(quantum minimum) - E(classical minimum) > 0 Classically, we can calculate the natural oscillation frequency that the particle would have if we were to give it a small push. Quantum mechanically, it is now an undergraduate exercise to use the Heisenberg uncertainty relation (more precisely, Schroedinger's differential equation) to show that. E(ZPE) approx. = (1/2) hbar omega0 where hbar is Planck's constant times and omega0 is the natural oscillation frequency. The ZPE in this sense shows up almost everywhere: it affects molecular bonds, condensed matter physics, small oscillations of any system. The next step is to realize that the electromagnetic field can be thought of as an infinite collection of coupled oscillators--one at each point in space. Again, the classical ground state is the case in which the electric and magnetic fields both must be zero. Quantum effects mean that this case does not hold true; there is also a Heisenberg uncertainty principle for electric and magnetic fields (it's a little more complex). The good news is that the potential for electromagnetism is exactly quadratic and so can be solved exactly. The bad news is that there is an infinite number of modes. Formally we can write (Electromagnetic vacuum energy) = sum over all modes (1/2) hbar omega(mode) The infinity in this equation is what excites the free lunch crowd (the modern descendants of the perpetual motion crowd), who envision an endless ZPE for humanity to tap into. Not quite, unfortunately..The first and most obvious problem is that there are other quantum fields in the universe apart from electromagnetism. Electrons, for starters, plus neutrinos, quarks, gluons, W, Z, Higgs and so on. In particular, if you do the calculation for electrons you will find that what are known as Fermi statistics give rise to an extra minus sign in the calculation. Adding minus infinity to plus infinity gives mathematicians nightmares and even makes theoretical physicists worry a little. Fortunately, nature does not worry about what the mathematicians or physicists think and does the job for us automatically. Consider the grand total vacuum energy (once we have added in all quantum fields, all particle interactions, kept everything finite by hook or by crook, and taken all the proper limits at the end of the day). This grand total vacuum energy has another name: it is called the "cosmological constant," and it is something that we can measure observationally. In its original incarnation, the cosmological constant was something that Einstein put into General Relativity (his theory of gravity) by hand. Particle physicists have since taken over this idea and appropriated it for their own by giving it this more physical description in terms of the ZPE and the vacuum energy. Astrophysicists are now busy putting observational limits on the cosmological constant. From the cosmological point of view these limits are still pretty broad: the cosmological constant could potentially provide up to 60 percent to 80 percent of the total mass of the universe. From a particle physics point of view, however, these limits are extremely stringent: the cosmological constant is more than 10(-123) times smaller than one would naively estimate from particle physics equations. The cosmological constant could quite plausibly be exactly zero. (Physicists are still arguing on this point.) Even if the cosmological constant is not zero it is certainly small on a particle-physics scale, small on a human-engineering scale, and too tiny to be any plausible source of energy for human needs--not that we have any good ideas on how to accomplish large-scale manipulations of the cosmological constant anyway.Putting the more exotic fantasies of the free lunch crowd aside, is there anything more plausible that we could use the ZPE for? It turns out that small-scale manipulations of the ZPE are indeed possible. By introducing a conductor or a dielectric, one can affect the electromagnetic field and thus induce changes in the quantum mechanical vacuum, leading to changes in the ZPE. This is what underlies a peculiar physical phenomenon called the Casimir effect. In a classical world, perfectly neutral conductors do not attract one another. In a quantum world, however, the neutral conductors disturb the quantum electromagnetic vacuum and produce finite measurable changes in the energy as the conductors move around. Sometimes we can even calculate the change in energy and compare it with experiment. These effects are all undoubtedly real and uncontroversial but tiny. More controversial is the suggestion, made by the physicist Julian Schwinger, that the ZPE in dielectrics has something to do with sonoluminescence. The jury is still out on this one and there is a lot of polite discussion going on (both among experimentalists, who are unsure of which of the competing mechanisms is the correct one, and among theorists, who still disagree on the precise size and nature of the Casimir effect in dielectrics.) Even more speculative is the suggestion that relates the Casimir effect to "starquakes" on neutron stars and to gamma ray bursts. In summary, there is no doubt that the ZPE, vacuum energy and Casimir effect are physically real. Our ability to manipulate these quantities is limited but in some cases technologically interesting. But the free-lunch crowd has greatly exaggerated the importance of the ZPE. Notions of mining the ZPE should therefore be treated with extreme skepticism. From the way some enthusiasts talk about the zero-point energy, one might think that unlimited power is lying all around just waiting to be harnessed. Like many ideas that seem too good to be true, this one falls apart on closer examination, although the concept of the zero-point energy is quite fascinating in and of itself. John Obienin, a materials science researcher at the University of Nebraska at Omaha, explains: "Zero-point energy refers to random quantum fluctuations of the electromagnetic (and other) force fields that are present everywhere in the vacuum; in other words, an 'empty' vacuum is actually a seething cauldron of energy. This energy is present even at absolute zero temperature (-273 Celsius),and of course, even when no matter is present. The effect of these vacuum fields has been detected just barely--the effect is very tiny--by the attraction they induce in a capacitor, which is really just two close parallel metal plates. This effect is the famous prediction of Hendrick B. G. Casimir (made in 1948); it was very crudely 'confirmed' experimentally by M. J. Sparnaay in 1958. A recent, widely noted experiment by Steven K. Lamoreaux (Physical Review Letters, Vol. 78, No.1, pages. 5-8; January 6, 1997) gave a very precise and unambiguous confirmation of the existence of the Casimir force. "These vacuum fluctuations may have effects, both subtle and gross, on the behavior of microscopic particles and on the world around us. Russian physicist Andrei Sakharov speculated that they may give rise to the force of gravity. At present, nobody knows how to exploit the zero-point energy in a macroscopic device that delivers sizable amounts of energy. There is, however, a considerable fringe element (similar to those attracted to UFOs, astrology, numerology and so on) of people who speculate and fantasize about the possibility of exploiting the zero-point energy to achieve various technical marvels and the long-sought 'perpetual motion.' Consider yourself warned." John Baez is a member of the mathematics faculty at the University of California at Riverside and one of the moderators of the on-line sci.physics.research newsgroup. He adds some context: "The concept of vacuum energy shows up in certain computations in quantum field theory, which is the tool we use to conduct modern particle physics. In reality, particles interact with one another through a variety of forces. This is a complicated business, so in quantum-field theory we start by studying an idealized model in which particles do not interact at all. This is called a 'free-field theory.' Then we use this free-field theory as the basis for studying the 'interacting-field theory' we are really interested in. "In quantum-field theory, the vacuum state is defined to be the state having the least energy density. Something funny happens when we use a free-field theory to study an interacting-field theory: the vacuum state of the free-field theory is different from vacuum state of the interacting-field theory. The vacuum state of the interacting-field theory may have more or less energy than that of the free-field theory; the difference is called the vacuum energy. "One should not take this vacuum energy too literally, however, because the free-field theory is just a mathematical tool to help us understand what we are really interested in: the interacting theory. Only the interacting theory is supposed to correspond directly to reality. Because the vacuum state of the interacting theory is the state of least energy in reality, there is no way to extract the vacuum energy and use it for anything. "It is a bit like this: say a bank found it more convenient (for some strange reason) to start counting at 1,000, so that even when you had no money in the bank, your account read $1,000. You might get excited and try to spend this $1,000, but the bank would say, 'Sorry, that $1,000 is just an artifact of how we do our bookkeeping: you're actually flat broke.' "Similarly, one should not get one's hope up when people talk about vacuum energy. It is just how we do our bookkeeping in quantum field theory. There is much more to say about why we do our bookkeeping this funny way, but I will stop here." Paul A. Deck, assistant professor of chemistry at Virginia Polytechnic Institute and State University, gives a chemical perspective on this question: "The zero-point energy cannot be harnessed in the traditional sense. The idea of zero-point energy is that there is a finite, minimum amount of motion (more accurately, kinetic energy) in all matter, even at absolute zero. For example, chemical bonds continue to vibrate in predictable ways. But releasing the energy of this motion is impossible, because then the molecule would be left with less than the minimum amount that the laws of quantum physics require it to have."
~Herr von Bradford
— with Upendra YadavThe Zero Point Energy (ZPE) is an intrinsic and unavoidable part of quantum physics. The ZPE has been studied, both theoretically and experimentally, since the discovery of quantum mechanics in the 1920s and there can be no doubt that the ZPE is a real physical effect. The "vacuum energy" is a specific example of ZPE which has generated considerable doubt and confusion. In a completely empty flat universe, calculations of the vacuum energy yield infinite values of both positive and negative sign--something that obviously does not correspond to the nature of the real world. Observation indicates that in our universe the grand total vacuum energy is extremely small and quite possibly exactly zero. Many theorists suspect that the total vacuum energy is exactly zero. It definitely is possible to manipulate the vacuum energy. Any objects that change the vacuum energy (electrical conductors, dielectrics and gravitational fields, for instance) distort the quantum mechanical vacuum state. These changes in the vacuum energy are often easier to calculate than the total vacuum energy itself. Sometimes we can even measure these changes in the vacuum energy in laboratory experiments. In classical physics, if you have a particle that is acted on by some conservative force, the total energy is E = (1/2) mv2 + V(x). To find the classical ground state, set the velocity to zero to minimize the kinetic energy, (1/2)m v2, and put the particle at the point where it has the lowest potential energy V(x). But this result is only a classical approximation to the real world. Because the classical ground state completely specifies both the particle's speed (zero) and position (at the minimum), it violates the famous Heisenberg Uncertainty Principle (m dv dx > hbar). Quantum physics, via the Uncertainty Principle, forces the particle to spread out both in position and velocity and so causes it to have an energy somewhat higher than the classical minimum. The ZPE is defined as this shift: E(ZPE) = E(quantum minimum) - E(classical minimum) > 0 Classically, we can calculate the natural oscillation frequency that the particle would have if we were to give it a small push. Quantum mechanically, it is now an undergraduate exercise to use the Heisenberg uncertainty relation (more precisely, Schroedinger's differential equation) to show that. E(ZPE) approx. = (1/2) hbar omega0 where hbar is Planck's constant times and omega0 is the natural oscillation frequency. The ZPE in this sense shows up almost everywhere: it affects molecular bonds, condensed matter physics, small oscillations of any system. The next step is to realize that the electromagnetic field can be thought of as an infinite collection of coupled oscillators--one at each point in space. Again, the classical ground state is the case in which the electric and magnetic fields both must be zero. Quantum effects mean that this case does not hold true; there is also a Heisenberg uncertainty principle for electric and magnetic fields (it's a little more complex). The good news is that the potential for electromagnetism is exactly quadratic and so can be solved exactly. The bad news is that there is an infinite number of modes. Formally we can write (Electromagnetic vacuum energy) = sum over all modes (1/2) hbar omega(mode) The infinity in this equation is what excites the free lunch crowd (the modern descendants of the perpetual motion crowd), who envision an endless ZPE for humanity to tap into. Not quite, unfortunately..The first and most obvious problem is that there are other quantum fields in the universe apart from electromagnetism. Electrons, for starters, plus neutrinos, quarks, gluons, W, Z, Higgs and so on. In particular, if you do the calculation for electrons you will find that what are known as Fermi statistics give rise to an extra minus sign in the calculation. Adding minus infinity to plus infinity gives mathematicians nightmares and even makes theoretical physicists worry a little. Fortunately, nature does not worry about what the mathematicians or physicists think and does the job for us automatically. Consider the grand total vacuum energy (once we have added in all quantum fields, all particle interactions, kept everything finite by hook or by crook, and taken all the proper limits at the end of the day). This grand total vacuum energy has another name: it is called the "cosmological constant," and it is something that we can measure observationally. In its original incarnation, the cosmological constant was something that Einstein put into General Relativity (his theory of gravity) by hand. Particle physicists have since taken over this idea and appropriated it for their own by giving it this more physical description in terms of the ZPE and the vacuum energy. Astrophysicists are now busy putting observational limits on the cosmological constant. From the cosmological point of view these limits are still pretty broad: the cosmological constant could potentially provide up to 60 percent to 80 percent of the total mass of the universe. From a particle physics point of view, however, these limits are extremely stringent: the cosmological constant is more than 10(-123) times smaller than one would naively estimate from particle physics equations. The cosmological constant could quite plausibly be exactly zero. (Physicists are still arguing on this point.) Even if the cosmological constant is not zero it is certainly small on a particle-physics scale, small on a human-engineering scale, and too tiny to be any plausible source of energy for human needs--not that we have any good ideas on how to accomplish large-scale manipulations of the cosmological constant anyway.Putting the more exotic fantasies of the free lunch crowd aside, is there anything more plausible that we could use the ZPE for? It turns out that small-scale manipulations of the ZPE are indeed possible. By introducing a conductor or a dielectric, one can affect the electromagnetic field and thus induce changes in the quantum mechanical vacuum, leading to changes in the ZPE. This is what underlies a peculiar physical phenomenon called the Casimir effect. In a classical world, perfectly neutral conductors do not attract one another. In a quantum world, however, the neutral conductors disturb the quantum electromagnetic vacuum and produce finite measurable changes in the energy as the conductors move around. Sometimes we can even calculate the change in energy and compare it with experiment. These effects are all undoubtedly real and uncontroversial but tiny. More controversial is the suggestion, made by the physicist Julian Schwinger, that the ZPE in dielectrics has something to do with sonoluminescence. The jury is still out on this one and there is a lot of polite discussion going on (both among experimentalists, who are unsure of which of the competing mechanisms is the correct one, and among theorists, who still disagree on the precise size and nature of the Casimir effect in dielectrics.) Even more speculative is the suggestion that relates the Casimir effect to "starquakes" on neutron stars and to gamma ray bursts. In summary, there is no doubt that the ZPE, vacuum energy and Casimir effect are physically real. Our ability to manipulate these quantities is limited but in some cases technologically interesting. But the free-lunch crowd has greatly exaggerated the importance of the ZPE. Notions of mining the ZPE should therefore be treated with extreme skepticism. From the way some enthusiasts talk about the zero-point energy, one might think that unlimited power is lying all around just waiting to be harnessed. Like many ideas that seem too good to be true, this one falls apart on closer examination, although the concept of the zero-point energy is quite fascinating in and of itself. John Obienin, a materials science researcher at the University of Nebraska at Omaha, explains: "Zero-point energy refers to random quantum fluctuations of the electromagnetic (and other) force fields that are present everywhere in the vacuum; in other words, an 'empty' vacuum is actually a seething cauldron of energy. This energy is present even at absolute zero temperature (-273 Celsius),and of course, even when no matter is present. The effect of these vacuum fields has been detected just barely--the effect is very tiny--by the attraction they induce in a capacitor, which is really just two close parallel metal plates. This effect is the famous prediction of Hendrick B. G. Casimir (made in 1948); it was very crudely 'confirmed' experimentally by M. J. Sparnaay in 1958. A recent, widely noted experiment by Steven K. Lamoreaux (Physical Review Letters, Vol. 78, No.1, pages. 5-8; January 6, 1997) gave a very precise and unambiguous confirmation of the existence of the Casimir force. "These vacuum fluctuations may have effects, both subtle and gross, on the behavior of microscopic particles and on the world around us. Russian physicist Andrei Sakharov speculated that they may give rise to the force of gravity. At present, nobody knows how to exploit the zero-point energy in a macroscopic device that delivers sizable amounts of energy. There is, however, a considerable fringe element (similar to those attracted to UFOs, astrology, numerology and so on) of people who speculate and fantasize about the possibility of exploiting the zero-point energy to achieve various technical marvels and the long-sought 'perpetual motion.' Consider yourself warned." John Baez is a member of the mathematics faculty at the University of California at Riverside and one of the moderators of the on-line sci.physics.research newsgroup. He adds some context: "The concept of vacuum energy shows up in certain computations in quantum field theory, which is the tool we use to conduct modern particle physics. In reality, particles interact with one another through a variety of forces. This is a complicated business, so in quantum-field theory we start by studying an idealized model in which particles do not interact at all. This is called a 'free-field theory.' Then we use this free-field theory as the basis for studying the 'interacting-field theory' we are really interested in. "In quantum-field theory, the vacuum state is defined to be the state having the least energy density. Something funny happens when we use a free-field theory to study an interacting-field theory: the vacuum state of the free-field theory is different from vacuum state of the interacting-field theory. The vacuum state of the interacting-field theory may have more or less energy than that of the free-field theory; the difference is called the vacuum energy. "One should not take this vacuum energy too literally, however, because the free-field theory is just a mathematical tool to help us understand what we are really interested in: the interacting theory. Only the interacting theory is supposed to correspond directly to reality. Because the vacuum state of the interacting theory is the state of least energy in reality, there is no way to extract the vacuum energy and use it for anything. "It is a bit like this: say a bank found it more convenient (for some strange reason) to start counting at 1,000, so that even when you had no money in the bank, your account read $1,000. You might get excited and try to spend this $1,000, but the bank would say, 'Sorry, that $1,000 is just an artifact of how we do our bookkeeping: you're actually flat broke.' "Similarly, one should not get one's hope up when people talk about vacuum energy. It is just how we do our bookkeeping in quantum field theory. There is much more to say about why we do our bookkeeping this funny way, but I will stop here." Paul A. Deck, assistant professor of chemistry at Virginia Polytechnic Institute and State University, gives a chemical perspective on this question: "The zero-point energy cannot be harnessed in the traditional sense. The idea of zero-point energy is that there is a finite, minimum amount of motion (more accurately, kinetic energy) in all matter, even at absolute zero. For example, chemical bonds continue to vibrate in predictable ways. But releasing the energy of this motion is impossible, because then the molecule would be left with less than the minimum amount that the laws of quantum physics require it to have."
~Herr von Bradford
Conservation of linear momentum
The conservation of momentum is a fundamental concept of physics along with the conservation of energy and the conservation of mass. Momentum is defined to be the mass of an object multiplied by the velocity of the object. The conservation of momentum states that, within some problem domain, the amount of momentum remains constant; momentum is neither created nor destroyed, but only changed through the action of forces as described by Newton's laws of motion. Dealing with momentum is more difficult than dealing with mass and energy because momentum is a vector quantity having both a magnitude and a direction. Momentum is conserved in all three physical directions at the same time. It is even more difficult when dealing with a gas because forces in one direction can affect the momentum in another direction because of the collisions of many molecules. On this slide, we will present a very, very simplified flow problem where properties only change in one direction. The problem is further simplified by considering a steady flow which does not change with time and by limiting the forces to only those associated with the pressure. Be aware that real flow problems are much more complex than this simple example.
Let us consider the flow of a gas through a domain in which flow properties only change in one direction, which we will call "x". The gas enters the domain at station 1 with some velocity u and some pressure p and exits at station 2 with a different value of velocity and pressure. For simplicity, we will assume that the density r remains constant within the domain and that the area A through which the gas flows also remains constant. The location of stations 1 and 2 are separated by a distance called del x. (Delta is the little triangle on the slide and is the Greek letter "d". Mathematicians often use this symbol to denote a change or variation of a quantity. The web print font does not support the Greek letters, so we will just call it "del".) A change with distance is referred to as a gradient to avoid confusion with a change with time which is called a rate. The velocity gradient is indicated by del u / del x; the change in velocity per change in distance. So at station 2, the velocity is given by the velocity at 1 plus the gradient times the distance.
u2 = u1 + (del u / del x) * del x
A similar expression gives the pressure at the exit:
p2 = p1 + (del p / del x) * del x
Newton's second law of motion states that force F is equal to the change in momentum with respect to time. For an object with constant mass m this reduces to the mass times acceleration a. An acceleration is a change in velocity with a change in time (del u / del t). Then:
F = m * a = m * (del u / del t)
The force in this problem comes from the pressure gradient. Since pressure is a force per unit area, the net force on our fluid domain is the pressure times the area at the exit minus the pressure times the area at the entrance.
F = - [(p * A)2 - (p * A)1] = m * [(u2 - u1) / del t]
The minus sign at the beginning of this expression is used because gases move from a region of high pressure to a region of low pressure; if the pressure increases with x, the velocity will decrease. Substituting for our expressions for velocity and pressure:
- [{(p + (del p / del x) * del x} * A) - (p * A)] = m * [(u + (del u / del x) * del x - u) / del t]
Simplify:
- (del p / del x) * del x * A = m * (del u / del x) * del x / del t
Noting that (del x / del t) is the velocity and that the mass is the density r times the volume (area times del x):
- (del p / del x) * del x * A = r * del x * A * (del u / del x) * u
Simplify:
- (del p / del x) = r * u * (del u / del x)
The del p / del x and del u / del x represent the pressure and velocity gradients. If we shrink our domain down to differential sizes, these gradients become differentials:
- dp/dx = r * u * du/dx
This is a one dimensional, steady form of Euler's Equation. It is interesting to note that the pressure drop of a fluid (the term on the left) is proportional to both the value of the velocity and the gradient of the velocity. A solution of this momentum equation gives us the form of the dynamic pressure that appears in Bernoulli's Equation.
SOURCE: http://www.grc.nasa.gov/
~Herr von Bradford
Let us consider the flow of a gas through a domain in which flow properties only change in one direction, which we will call "x". The gas enters the domain at station 1 with some velocity u and some pressure p and exits at station 2 with a different value of velocity and pressure. For simplicity, we will assume that the density r remains constant within the domain and that the area A through which the gas flows also remains constant. The location of stations 1 and 2 are separated by a distance called del x. (Delta is the little triangle on the slide and is the Greek letter "d". Mathematicians often use this symbol to denote a change or variation of a quantity. The web print font does not support the Greek letters, so we will just call it "del".) A change with distance is referred to as a gradient to avoid confusion with a change with time which is called a rate. The velocity gradient is indicated by del u / del x; the change in velocity per change in distance. So at station 2, the velocity is given by the velocity at 1 plus the gradient times the distance.
u2 = u1 + (del u / del x) * del x
A similar expression gives the pressure at the exit:
p2 = p1 + (del p / del x) * del x
Newton's second law of motion states that force F is equal to the change in momentum with respect to time. For an object with constant mass m this reduces to the mass times acceleration a. An acceleration is a change in velocity with a change in time (del u / del t). Then:
F = m * a = m * (del u / del t)
The force in this problem comes from the pressure gradient. Since pressure is a force per unit area, the net force on our fluid domain is the pressure times the area at the exit minus the pressure times the area at the entrance.
F = - [(p * A)2 - (p * A)1] = m * [(u2 - u1) / del t]
The minus sign at the beginning of this expression is used because gases move from a region of high pressure to a region of low pressure; if the pressure increases with x, the velocity will decrease. Substituting for our expressions for velocity and pressure:
- [{(p + (del p / del x) * del x} * A) - (p * A)] = m * [(u + (del u / del x) * del x - u) / del t]
Simplify:
- (del p / del x) * del x * A = m * (del u / del x) * del x / del t
Noting that (del x / del t) is the velocity and that the mass is the density r times the volume (area times del x):
- (del p / del x) * del x * A = r * del x * A * (del u / del x) * u
Simplify:
- (del p / del x) = r * u * (del u / del x)
The del p / del x and del u / del x represent the pressure and velocity gradients. If we shrink our domain down to differential sizes, these gradients become differentials:
- dp/dx = r * u * du/dx
This is a one dimensional, steady form of Euler's Equation. It is interesting to note that the pressure drop of a fluid (the term on the left) is proportional to both the value of the velocity and the gradient of the velocity. A solution of this momentum equation gives us the form of the dynamic pressure that appears in Bernoulli's Equation.
SOURCE: http://www.grc.nasa.gov/
~Herr von Bradford
Sunday, June 15, 2014
NET Physics Coaching - KAS Institute
KAS Institute, 11th Mile Jn., Cherthala 688539
Build your career skills and experience at KAS
JRF/ NET coaching program in Physics for current students, recent post-graduates, repeaters and working professionals.
The select coaching options at KAS Institute provide unique opportunities for current students and repeaters to develop their Physics acumen. Students from all over Kerala expand their skills in Physics through KAS
Our coaching schedule suits the requirement of many a student
Classes on Saturday, Sunday and Holidays.
Hostel facility for girl students
Professionals and students can master Advances in Physics while strengthening their strategic thinking skills.
Next batch starts on July 6, 2014
PH; 9142020185
Mail: kasolaman@gmail.com
Monday, June 09, 2014
KAS Institute- CSIR-JRF/NET Physical Science
CSIR-JRF/NET Physical Science Next Batch starts at KAS Institute, 11th-Mile Jn, Cherthala on July 6, 2014
CSIR-UGC-JRF-NET 2013 DEC QN PAPER SOLNPh: 9142020185
Sunday, June 01, 2014
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