SR Exercise
in [Relativity]
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From: Tony M on 26 Apr 2010 11:30 On Apr 26, 10:37 am, harald <h...(a)swissonline.ch> wrote: > On Apr 25, 7:54 pm, Tony M <marc...(a)gmail.com> wrote: > > > > > > > On Apr 23, 12:55 pm, PD <thedraperfam...(a)gmail.com> wrote: > > > > On Apr 23, 7:49 am, Tony M <marc...(a)gmail.com> wrote: > > > > > A spaceship with a mass of 1000kg is initially at rest relative to an > > > > external observer. The spaceship emits one photon with an energy of > > > > 1kg x c^2 as measured by the observer. We define a closed system > > > > including the spaceship and the photon. Please provide the values for: > > > > -invariant mass of the system > > > > -total energy of the system > > > > -velocity of the center of mass, and momentum of the system as > > > > measured by the observer > > > > -velocity of spaceship as measured by the observer > > > > -rest mass of the spaceship (kg, minimum 4 decimals) > > > > We'll take c to be 3E8m/s exactly, just to make the calculations > > > simpler, rather than to generate precise numbers. The purpose here is > > > to show how the calculations work. > > > The system energy is 9E19 J, and the invariant mass of the system is > > > 1000 kg. > > > The energy of the emitted photon is 9E16 J, and its momentum is 3E8 > > > kg*m/s, and its rest mass is zero. The rocket's momentum is therefore > > > 3E8 kg*m/s in the opposite direction. The momentum (and velocity) of > > > the center of mass is zero. > > > The total energy of the rocket is 8.991E19 J. It's rest mass after > > > emission of the photon is therefore given by m^2 = (E/c^2)^2 - (p/ > > > c)^2, which yields m = 998.9995 kg. > > > The speed of the rocket is given by beta = (v/c) = pc/E = 0.0010, so > > > the rocket is traveling at 3E5 m/s in this frame. > > > > You'll notice the sum of the rest masses does not add up to 1000 kg. > > > > PD > > > Well PD, you're the only one to give the complete answer and I came up > > with the same numbers. If you look at the 998.9995 value you will see > > my reason in asking for 4 decimals. Now, if I can get you to expand > > your understanding of mass and accept relativistic mass and photon > > mass as very real, you're all set. We can give dlzc partial credit for > > his answer as well. > > > So, let's analyze that invariant mass of the system. Why did you say > > it's 1000kg? Is it just because the term invariant means it doesn't > > change or is there more to it? Since the only rest mass left in the > > system is the 998.9995kg how do we come up with 1000kg invariant mass? > > You told us that the invariant mass is not the sum of the rest masses. > > What is it then? Where is that extra mass coming from and what kind is > > it? Could the invariant mass be the sum of the relativistic masses as > > measured by the above observer? Would that mean the relativistic mass > > and the invariant mass are equally real? Not to mention that we're > > also including the mass of the photon, which is supposed to have no > > mass. > > > The invariant mass of a system is the lowest mass an observer can > > measure for that system. Not coincidentally, this happens when the > > observer is in the frame in which the system has no momentum, when the > > observer is at rest with the center of mass of the system. The mass > > the observer will measure in this case is the sum of the total > > energies divided by c^2, for all particles, including photons. And > > what does the total energy of a particle divided by c^2 represent? > > It's nothing other than it's relativistic mass. So the invariant mass > > of a system, which is a real, fundamental, measurable and accepted > > mass is if fact a sum of relativistic masses (in a particular frame of > > reference). Doesn't this mean that the invariant mass is only as real > > as the relativistic mass and vice-versa? > > > We also notice the invariant mass is the same before and after the > > emission of the photon. How is that possible? We have just produced > > 1kg x c^2 worth of energy; weren't we supposed to "consume" an > > equivalent amount of mass? A lot of people here claim that mass gets > > converted to energy. One guy actually said the mass of the Universe is > > decreasing because stars convert mass to energy. How come the > > invariant mass is the same before and after, and it will remain > > constant for that closed system no matter what happens inside, > > regardless how many photons we emit, or whether we have fission, > > fusion or matter/antimatter annihilations taking place? Does this mean > > that the invariant mass of a closed system is conserved, in the same > > way the energy is conserved? Should we also conclude that photons have > > real mass which contributes to the mass of a system, and not just > > "energy which could be converted to an equivalent amount of mass"? > > > Or should we just take the easy way out and simply deny the invariant > > mass along with the relativistic mass, and say that mass does not > > apply to a system, like one of our friends here? > > Funny enough, in 1905 Einstein phrased his words very carefully and > more correctly than many modern publications: > > "If a body gives off the energy L in the form of radiation, its mass > diminishes by L/c². [...] the energy withdrawn from the body becomes > energy of radiation [...] > The mass of a body is a measure of its energy-content" > -http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ > > See also the FAQ:http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.... > > Harald- Hide quoted text - > > - Show quoted text - Right, as you can see from PD's answer the mass of the spaceship decreases but the mass of the whole system, which includes the photon, remains unchanged.
From: Androcles on 26 Apr 2010 12:29 "Tony M" <marcuac(a)gmail.com> wrote in message news:5f1e8db1-ecf2-4e12-88be-cd5d06c44614(a)12g2000yqi.googlegroups.com... On Apr 26, 10:59 am, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote: > "harald" <h...(a)swissonline.ch> wrote in message > > news:14b34d19-83e5-45ba-86e0-095a04d2c64e(a)s9g2000yqa.googlegroups.com... > On Apr 25, 7:54 pm, Tony M <marc...(a)gmail.com> wrote: > > > > > > > On Apr 23, 12:55 pm, PD <thedraperfam...(a)gmail.com> wrote: > > > > On Apr 23, 7:49 am, Tony M <marc...(a)gmail.com> wrote: > > > > > A spaceship with a mass of 1000kg is initially at rest relative to > > > > an > > > > external observer. The spaceship emits one photon with an energy of > > > > 1kg x c^2 as measured by the observer. We define a closed system > > > > including the spaceship and the photon. Please provide the values > > > > for: > > > > -invariant mass of the system > > > > -total energy of the system > > > > -velocity of the center of mass, and momentum of the system as > > > > measured by the observer > > > > -velocity of spaceship as measured by the observer > > > > -rest mass of the spaceship (kg, minimum 4 decimals) > > > > We'll take c to be 3E8m/s exactly, just to make the calculations > > > simpler, rather than to generate precise numbers. The purpose here is > > > to show how the calculations work. > > > The system energy is 9E19 J, and the invariant mass of the system is > > > 1000 kg. > > > The energy of the emitted photon is 9E16 J, and its momentum is 3E8 > > > kg*m/s, and its rest mass is zero. The rocket's momentum is therefore > > > 3E8 kg*m/s in the opposite direction. The momentum (and velocity) of > > > the center of mass is zero. > > > The total energy of the rocket is 8.991E19 J. It's rest mass after > > > emission of the photon is therefore given by m^2 = (E/c^2)^2 - (p/ > > > c)^2, which yields m = 998.9995 kg. > > > The speed of the rocket is given by beta = (v/c) = pc/E = 0.0010, so > > > the rocket is traveling at 3E5 m/s in this frame. > > > > You'll notice the sum of the rest masses does not add up to 1000 kg. > > > > PD > > > Well PD, you're the only one to give the complete answer and I came up > > with the same numbers. If you look at the 998.9995 value you will see > > my reason in asking for 4 decimals. Now, if I can get you to expand > > your understanding of mass and accept relativistic mass and photon > > mass as very real, you're all set. We can give dlzc partial credit for > > his answer as well. > > > So, let's analyze that invariant mass of the system. Why did you say > > it's 1000kg? Is it just because the term invariant means it doesn't > > change or is there more to it? Since the only rest mass left in the > > system is the 998.9995kg how do we come up with 1000kg invariant mass? > > You told us that the invariant mass is not the sum of the rest masses. > > What is it then? Where is that extra mass coming from and what kind is > > it? Could the invariant mass be the sum of the relativistic masses as > > measured by the above observer? Would that mean the relativistic mass > > and the invariant mass are equally real? Not to mention that we're > > also including the mass of the photon, which is supposed to have no > > mass. > > > The invariant mass of a system is the lowest mass an observer can > > measure for that system. Not coincidentally, this happens when the > > observer is in the frame in which the system has no momentum, when the > > observer is at rest with the center of mass of the system. The mass > > the observer will measure in this case is the sum of the total > > energies divided by c^2, for all particles, including photons. And > > what does the total energy of a particle divided by c^2 represent? > > It's nothing other than it's relativistic mass. So the invariant mass > > of a system, which is a real, fundamental, measurable and accepted > > mass is if fact a sum of relativistic masses (in a particular frame of > > reference). Doesn't this mean that the invariant mass is only as real > > as the relativistic mass and vice-versa? > > > We also notice the invariant mass is the same before and after the > > emission of the photon. How is that possible? We have just produced > > 1kg x c^2 worth of energy; weren't we supposed to "consume" an > > equivalent amount of mass? A lot of people here claim that mass gets > > converted to energy. One guy actually said the mass of the Universe is > > decreasing because stars convert mass to energy. How come the > > invariant mass is the same before and after, and it will remain > > constant for that closed system no matter what happens inside, > > regardless how many photons we emit, or whether we have fission, > > fusion or matter/antimatter annihilations taking place? Does this mean > > that the invariant mass of a closed system is conserved, in the same > > way the energy is conserved? Should we also conclude that photons have > > real mass which contributes to the mass of a system, and not just > > "energy which could be converted to an equivalent amount of mass"? > > > Or should we just take the easy way out and simply deny the invariant > > mass along with the relativistic mass, and say that mass does not > > apply to a system, like one of our friends here? > > Funny enough, in 1905 Einstein phrased his words very carefully and > more correctly than many modern publications: > > "If a body gives off the energy L in the form of radiation, its mass > diminishes by L/c�. [...] the energy withdrawn from the body becomes > energy of radiation [...] > The mass of a body is a measure of its energy-content" > -http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ > > See also the > FAQ:http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.... > > Harald > > Tony M's scenario is impossible, the spaceship emits one photon with an > energy of 1kg x c^2 as measured by the observer and another photon > going in the opposite direction with an energy of 1kg x c^2 not measured > and not seen by the observer, as per conservation of momentum. > The energy of the two photons is of course 1/2mc^2 each. Having made > that error to begin with the rest of his sums are WRONG.- Hide quoted > text - > > - Show quoted text - What second photon? ==================================================== The one you can't see that I've already told you about. Not even Einstein would claim the Sun doesn't shine on the opposite side. For every photon there is and equal an opposite rephoton. Phased array radar shoots narrow beams of radiation (aka a photon stream). http://www.androcles01.pwp.blueyonder.co.uk/rephoton.gif ==================================================== There�s only one photon. Are you saying photons can only be emitted in pairs, going in opposite directions? ==================================================== Yes, photons can only be emitted in pairs. Of course you could always put a mirror behind the Sun... oh wait, a planet will do, they reflect photons. I'm quite certain the other side of the Sun also shines. Conservation of momentum applies just fine for the one photon and the spaceship. See PD�s answer which is correct for the given scenario. ==================================================== There are no one-photon scenarios. I'm not arguing with E=mc^2, nor am I arguing with KE = 1/2mv^2, they both fit Newtonian Mechanics perfectly when v = c and you have two photons. 2 * 1/2mv^2 = mv^2. What I am arguing with is your single photon. "I've lost interest. Foam and blather and waste all the time you want. You're not getting anywhere." -- Phuckwit Duck (Meaning "I lost that argument, those grapes are sour".) Ref: d23006a4-4a88-4efb-b1f4-12b11539952c(a)c34g2000yqn.googlegroups.com "You are not entitled to be educated. Someone who insists on being willfully ignorant does not deserve to be dissuaded. Nobody owes you anything. Nobody *should* do anything for you. It's your choice to learn or not to learn."-- Phuckwit Duck Ref: 571b8ace-cca8-4392-ba69-0a328320ad62(a)o28g2000yqh.googlegroups.com Phuckwit Duck is a moronic bigot anyway, so don't hold him up to support your wild conjecture of a single photon being possible.
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