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Author Topic: Is hot water heavier than cold water, and how can I prove this?  (Read 19100 times)

Offline thedoc

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michael clark asked the Naked Scientists:
   
How can i prove to my boss that hot water is heavier than cold water in such away that he can understand? As simple as possible please. Thank you
What do you think?
« Last Edit: 25/11/2014 15:30:04 by _system »


 

Offline alancalverd

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This is only true in the region 0  - 4C, so you will need a freezer to demonstrate it.

Fill a cup close to the brim with water, put  it in the freezer, and stir it with a thermometer. When it reaches about 5C, top it up to the brim, then watch what happens as it cools further. It will spill over the edge just before it freezes because cold water takes up more space (i.e. is less dense) than warm.

If you use a conical bottle, say a wine bottle, the effect will be more spectacular.   
 

Offline PmbPhy

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Quote from: alancalverd
This is only true in the region 0  - 4C, so you will need a freezer to demonstrate it.

Fill a cup close to the brim with water, put  it in the freezer, and stir it with a thermometer. When it reaches about 5C, top it up to the brim, then watch what happens as it cools further. It will spill over the edge just before it freezes because cold water takes up more space (i.e. is less dense) than warm.

If you use a conical bottle, say a wine bottle, the effect will be more spectacular.   
Depending on the exact meaning of what the OP's question really is, i.e. what he's seeking to learn, that's wrong. You appear to be addressing the question about the density of water and not the mass of the same amount of water. Water becomes denser when the temperature is between 0 and 4C as you said. That's a physical fact and its why ice doesn't form on the bottom on lakes but on the top.

However mass is different. According to Einstein's theory of relativity E =mc2 which means that given a finite amount of mass, when you add energy to it then it's mass will increase. However this increase is very very small. Too small in fact to measure in the laboratory.
 

Offline evan_au

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You could try it with a clear container and a jug of hot and cold water, each containing a different food dye.

Pour water from one jug into the clear container, and wait for circulation to almost stop. Then very carefully add water from the second container. There will be some mixing at the interface, but overall the less dense fluid will remain at the top.
Placing the more dense fluid on the top will cause more mixing.

But most temperatures that you try for this experiment (especially around "room temperature") will show that the cooler water is more dense.
 

Offline alancalverd

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However mass is different. According to Einstein's theory of relativity E =mc2 which means that given a finite amount of mass, when you add energy to it then it's mass will increase. However this increase is very very small. Too small in fact to measure in the laboratory.

Hey! I thought I was the King Nitpicker round here! So just to reassert my position

Quote
How can i prove to my boss

You can't get away with theoretical physics, Pete! This guy is asking for proof. So please let me have your calculation of the relativistic increase in molecular mass over the phase range of liquid water, and an indication of how he can measure it against the background of the variable density of the material. You have until nightfall (London time) to comply, otherwise .... off to the Tower with you, and may the executioner's blade be sharp!
 

Offline Roadrunner

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Will he believe that all scuba divers know that cold water sits on the bottom of a body of water, and warm water sits on top of the cold water?

Even on cloudy days - the New England Atlantic Ocean is one example of this, though this principle holds true in fresh water too. The water on top of a body of water can be 4 to 11 C, and the water at the bottom (all others variables being equal) will measure colder.

Tell him to scuba dive to find his "proof,"  or just pull out any oceanography book. It is a very basic fact that cold water is heavier than warm water.

Quote
But, cold water is more compressible than warm water. That is, it is easier to deform a cold parcel than a warm parcel. Therefore cold water becomes denser than warm water when they are both submerged to the same pressure.

newbielink:https://www.rsmas.miami.edu/personal/lbeal/MPO%20503/Lectures%203%264.html [nonactive]

That's about as simple as one can have it.
« Last Edit: 26/11/2014 21:15:34 by Roadrunner »
 

Offline lightarrow

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However mass is different. According to Einstein's theory of relativity E =mc2 which means that given a finite amount of mass, when you add energy to it then it's mass will increase.
...if that amount of mass is still, otherwise it's not true ( I know that you knows and I wouldn't give the idea that I'm correcting you because I believe you weren't aware of it 😊).

--
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Offline lightarrow

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So please let me have your calculation of the relativistic increase in molecular mass over the phase range of liquid water,
Sorry, what do you mean with "relativistic increase in molecular mass over the phase range of liquid water"?

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Offline CliffordK

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Ok,
So for a little applied math.

E=mc2
m = E/c2

Now, let's assume that due to global warming, the oceans on Earth have uniformly warmed by 1C (1 calorie/cc, or 4.184 joules/cc)

Ok, so there is about 1,350,000,000 km3 of water in the oceans.
10dm/m, 1000m/km, 10,000 dm/km
And one gets about:
1,350,000,000,000,000,000,000 dm3 of water, so about 1.35 x 1021 kg, or 1.35 x 1024 cc
And, raising the temperature by 1C (1 calorie/cc, or 4.184j/cc), one gets
5.65 x 1024 joules.

Now, for the speed of light, 299,792,458, and c2 = 9 x 1016 m2/s2.

So,
m/c2 = 5.65 x 1024 joules / 9x1016m2/s2.

And, you get that the oceans have increased in mass by about
62,800,000 kg

Whew, more than I expected.

---------------------------

Perhaps one should look at it in grams per cubic km of water.

So, one cubic km of water is: 1,000,000,000,000,000 (1x1015 cc)
And, raising 1 cubic km of water by 1C is about 4.184x1015 joules.

So,
4.184x1015 joules /  9x1016m2/s2 = 0.0465 kg/cubic km,

or about 46.5 grams/cubic km of water per degree Celsius.





« Last Edit: 27/11/2014 04:59:38 by CliffordK »
 

Offline jeffreyH

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Ok,
So for a little applied math.

E=mc2
m = E/c2

Now, let's assume that due to global warming, the oceans on Earth have uniformly warmed by 1C (1 calorie/cc, or 4.184 joules/cc)

Ok, so there is about 1,350,000,000 km3 of water in the oceans.
10dm/m, 1000m/km, 10,000 dm/km
And one gets about:
1,350,000,000,000,000,000,000 dm3 of water, so about 1.35 x 1021 kg, or 1.35 x 1024 cc
And, raising the temperature by 1C (1 calorie/cc, or 4.184j/cc), one gets
5.65 x 1024 joules.

Now, for the speed of light, 299,792,458, and c2 = 9 x 1016 m2/s2.

So,
m/c2 = 5.65 x 1024 joules / 9x1016m2/s2.

And, you get that the oceans have increased in mass by about
62,800,000 kg

Whew, more than I expected.

---------------------------

Perhaps one should look at it in grams per cubic km of water.

So, one cubic km of water is: 1,000,000,000,000,000 (1x1015 cc)
And, raising 1 cubic km of water by 1C is about 4.184x1015 joules.

So,
4.184x1015 joules /  9x1016m2/s2 = 0.0465 kg/cubic km,

or about 46.5 grams/cubic km of water per degree Celsius.

Use what he said that should impress the boss and scare the hell out of him thinking he is about to drown.
 

Offline alancalverd

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Except that the water is  hotter. This means that the molecules have more kinetic energy, not that they have gained in mass - you can't have your cake and eat it! What you have proved is that the energy required to raise the temperature of the oceans by 1 degree could be generated by the annihilation of 62.8 kilotons of matter. 
 

Offline PmbPhy

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Quote from: alancalverd
Except that the water is  hotter. This means that the molecules have more kinetic energy, not that they have gained in mass - you can't have your cake and eat it!
In Newtonian mechanics this is true. In relativistic mechanics its not true. However in the later case the increase is very very small. Too small in fact to be measured in a typical lab. One needs a mass spectrometer to measure such a small increase.

When the kinetic energy of a particle increases then so does it's mass since the particle is moving faster and mass depends on speed. Or to look at it another way (for those who abhor velocity dependent mass) The rest mass of liquid is a function of its energy content. When the liquid absorbs energy the kinetic energy of its atoms/molecules increases making them move faster. When they move faster the system of all the molecules has a greater inertia than when it was cooler. E.g. it weighs more and has greater momentum which in relativity goes into an increase in inertial mass.

I'm surprised you didn't know this alancalverd!? Or do you but neglected to mention it for some reason?
 

Offline alancalverd

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I was pointing out an error in Clifford's arithmetic. I would be grateful for your estimate of the mass gain from heating say a gram of water through 1 deg C.

The momentum argument doesn't hold for more than one molecule as they are moving randomly so the net momentum is always zero, regardless of temperature. Momentum is a vector.   
 

Offline PmbPhy

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Quote from: alancalverd
I was pointing out an error in Clifford's arithmetic. I would be grateful for your estimate of the mass gain from heating say a gram of water through 1 deg C.
That's trivial since that's defined to be an amount of energy equal to one calorie which equals 4.148 J. Since E = mc2 we have m = E/c2 = (4.184 J)/(3x108)2 = 4.649x10-17 kg

Quote from: alancalverd
The momentum argument doesn't hold for more than one molecule as they are moving randomly so the net momentum is always zero, regardless of temperature. Momentum is a vector.
That is incorrect. I'm sorry but you misunderstood what I said. When I wrote
Quote
E.g. it weighs more and has greater momentum which in relativity goes into an increase in inertial mass.
it meant that if it was being weighed then it would weigh more after it was heated than it was before and when it was moving then its momentum would be greater after it was heated than before it was heated.

Also, I obviously know that momentum is a vector. In relativity the relevant momentum is the 4-momentum. Don't forget that I'm a physicist too. :)
« Last Edit: 28/11/2014 03:17:23 by PmbPhy »
 

Offline alancalverd

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Sorry, you have repeated Clifford's mistake. You have calculated the mass that would have to be annihilated to produce 1 calorie of energy.
 

Offline RD

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How can i prove to my boss that hot water is heavier than cold water ...

Generally hotter water is less dense than cooler water , but it's the other way round between 0oC to 4oC  ...

Quote from: lsbu.ac.uk


... in water below about 4 C, warmer water sinks whereas when above about 4 C, warmer water rises ..."
http://www1.lsbu.ac.uk/water/density_anomalies.html#density
« Last Edit: 28/11/2014 03:21:37 by RD »
 

Offline PmbPhy

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Quote from: alancalverd
Hey! I thought I was the King Nitpicker round here! So just to reassert my position

Quote
How can i prove to my boss

You can't get away with theoretical physics, Pete! This guy is asking for proof.
I just saw this. I missed it when it was first posted. When someone asks for proof it by no means experimental proof. One would have to specify "experimental proof" for that to be a fact.

The reason I can "get away with theoretical physics" is that one can provide poof theoretically in cases such as this.

Recall that we use math heavily in physics to derive results. Recall what a theorem is
Quote
A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.
All the results that we have in physics that aren't laws are theorems of this nature and have been "proved" in this way.

To do so means to apply logic. Start with one or more propositions. These propositions serve as the premises of the argument. One then applies the appropriate reasoning which leads to the final proposition that one was seeking to arrive at known as the conclusion. This is how theorems are said to be proved. Such conclusions are quite general. Otherwise for an experimental "proof", i.e. if one demonstrates something in the lab, then that's more of a confirmation of a proposition rather than a proof of it.

Here I'm assuming that you knew that theorems were used in physics as well as in math too, correct? This is obvious because the language of physics is math. They appear when one uses math to prove a physical result. For example; Newton's Shell Theorem and Poynting's theorem theorem are two such example

Here is a longer list of examples: http://en.wikipedia.org/wiki/Category:Physics_theorems

In the case E = mc2 is a physical theorem that was proved by Einstein in 1905. He assumed that in the inertial rest  frame S of an object an amount of radiation was emitted in opposite directions in equal amounts. The total amount of radiation emitted was L. Then the system was analyzed by a frame S' which was moving at a constant velocity relative to S where the speed of that frame moved very slowly with respect to the speed of light.

Einstein then showed that the kinetic energy of the body as measured in S' was reduced by the amount (1/2)(L/c2)v2. From this Einstein concluded that
Quote
If a body gives off the energy L in the form of radiation, its mass diminishes by the amount L/c2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that - the mass of a body is a measure of its energy content: if the energy changes by the amount L, the mass changes in the same sense by L/9x1020, the energy measured in ergs, and the mass in grammes.
For different but  similar detailed derivation of E = mc2 please see my website at
athttp://home.comcast.net/~peter.m.brown/sr/mass_energy_equiv.htm

Sorry, you have repeated Clifford's mistake. You have calculated the mass that would have to be annihilated to produce 1 calorie of energy.
Clifford made no mistake, you did. Nothing personal Alan. However I'm quite surprised by this given your statement that you're an experimental physicist.
« Last Edit: 28/11/2014 03:27:33 by PmbPhy »
 

Offline alancalverd

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Point conceded, I think.

Now the question revolves around the definition of "heavier" and "proof". And then we have to consider the range of validity of that proof. Whilst it is obviously true that  a single molecule gains mass as it gains velocity (but relative to what, in the case of an isolated molecule in space?), "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment.

Consider two isolated molecules, with different velocity vectors. How do we measure their mass? In principle (I'll grant you this bit is well outside the realm of experimental physics) the only way is to measure the gravitational force exerted on a test mass that is stationary with respect to each molecule. And to our (theorietical) astonishment, they both have the same mass.

Back to the ensemble. Even if the said boss is indeed conversant with relativistic physics, he will still need an exceptional grasp of the Gibbs energy of dynamic Voronoi polyhedra if he is going to accept a theoretical proof, and despite advances in chaos theory since I last worked on shortrange order in water, the numbers remain very vague and entirely experimental.

So we have to subtract the calculated relativistic contribution to the ensemble mass from the currently-incalculable density variation.

Once again, I have to ask you to do the calculation, on the assumption that the said boss will understand it. I'll be generous here and let you use RD's (experimental) graph, just this once, to check your answer.  And perhaps you will let us know, as an aside, how many angels can sit on the head of a pin!

Frankly, I think putting a wine bottle in a freezer is a lot easier. But then I'm an experimentalist.

PS - belated good Thanksgiving wishes. Thanks to the Pilgrim Fathers I now have plenty of wine bottles and an empty freezer!
« Last Edit: 28/11/2014 08:44:49 by alancalverd »
 

Offline PmbPhy

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Quote from: alancalverd
Point conceded, I think.
I'm delighted to hear you say that. It demonstrates just how well you understand the mass-energy relation and the scientific method.  My compliments! :)

Quote from: alancalverd
Now the question revolves around the definition of "heavier" and "proof".
Those terms are well known and as such there shouldn't be any confusion about it. However I'll take these as challenges and go on. First off the term weight is well defined as the force on the object due to gravity. To be precise, consider a scale which is at rest in a gravitational field (e.g. sitting on a bench top in a lab). We then place a container of water of it. Then the force exerted on the scale to the force of gravity pressing down on the scale. That force is defined as the "weight" of the object. Please see my derivation on this in my website at
http://home.comcast.net/~peter.m.brown/gr/weight_moving_body.htm

Regarding "proof" I already elaborated on that in my last post. I think it only applies to theorems but I'm still studying the philosophy of science. In the meantime please see
http://www.psychologytoday.com/blog/the-scientific-fundamentalist/200811/common-misconceptions-about-science-i-scientific-proof
Quote
Misconceptions about the nature and practice of science abound, and are sometimes even held by otherwise respectable practicing scientists themselves.  I have dispelled some of them (misconceptions, not scientists) in earlier posts (for example, that beauty is in the eye of the beholder, beauty is only skin-deep, and you cant judge a book by its cover).  Unfortunately, there are many other misconceptions about science.  One of the most common misconceptions concerns the so-called scientific proofs.  Contrary to popular belief, there is no such thing as a scientific proof.


http://en.wikipedia.org/wiki/Scientific_evidence#Concept_of_.22scientific_proof.22
Quote
While the phrase "scientific proof" is often used in the popular media,[13] many scientists have argued that there is really no such thing. For example, Karl Popper once wrote that "In the empirical sciences, which alone can furnish us with information about the world we live in, proofs do not occur, if we mean by 'proof' an argument which establishes once and for ever the truth of a theory,"[14] and Satoshi Kanazawa has argued that "Proofs exist only in mathematics and logic, not in science."[15]
This is why I only spoke of proof in the mathematical sense in my last post.



See also



Quote from: alancalverd
And then we have to consider the range of validity of that proof. Whilst it is obviously true that  a single molecule gains mass as it gains velocity (but relative to what,...
When an author speaks of the velocity or speed of an object in the context of relativity they're referring to the speed relative to a particular frame of reference. For example; if I said instead "the speed of the particle with respect to frame S" then that's very specific. However it has the same content as the other way since in this way I didn't say with respect to what S is defined relative to either. So I leave it out and assume that the reader knows what it means. Especially ones who appear educated in relativity as you appear to be.

Quote from: alancalverd
... "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment.
See again http://home.comcast.net/~peter.m.brown/gr/weight_moving_body.htm

Notice that in that page I refer to a container containing a large number of particles moving randomly. For the weight of the box to increase with an increase in energy then the weight of each and every particle must increase. Since the box contains a large number of particles moving randomly its safe to speak of temperature and an input of heat.

Quote from: alancalverd
Consider two isolated molecules, with different velocity vectors. How do we measure their mass?
Their relativistic mass does not depend on its direction so we only need to measure its mass as a function of speed. To do that one can measure the mass of a moving particle such as a molecule by ionizing it and using a cyclotron to measure its mass by determining it's deflection by a magnetic field and measuring its radius of orbit. See
http://home.comcast.net/~peter.m.brown/sr/cyclotron.htm

You're an experimental physicist aren't you? If so they why didn't you know this?

Quote from: alancalverd
Once again, I have to ask you to do the calculation, ...
I don't see the point. I'm going to assume that "michael clark" is satisfied.
« Last Edit: 28/11/2014 16:17:59 by PmbPhy »
 

Offline PmbPhy

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michael clark asked the Naked Scientists:
   
How can i prove to my boss that hot water is heavier than cold water in such away that he can understand? As simple as possible please. Thank you
What do you think?

See also Astronomy Without A Telescope Mass Is Energy
http://www.universetoday.com/91132/astronomy-without-a-telescope-mass-is-energy/
Quote
Similarly you might consider that your cup of coffee is more massive when its hot and gets measurably less massive when it cools down. Matter, in terms of protons, neutrons, electrons and coffee, is largely conserved throughout this process. But, for a while, the heat energy really does add to the mass of the system although since its a mass of m=e/c2, it is a very tiny amount of mass.
I'd like to note that the assertion in the title, Mass Is Energy, is wrong. Mass is most certainly not energy!! To explain why takes an entire thread/physics journal article. I can upload one to my website and post the link to it here if anybody seriously wants to read it? Mendel Sachs wrote a paper on this subject.
 

Offline lightarrow

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Generally hotter water is less dense than cooler water , but it's the other way round between 0C to 4C  ...

Quote from: lsbu.ac.uk

From that picture it seems that what you say happens not only between 0C to 4C but also between -40C to 4C...

--
lightarrow
 

Offline jeffreyH

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Point conceded, I think.

Now the question revolves around the definition of "heavier" and "proof". And then we have to consider the range of validity of that proof. Whilst it is obviously true that  a single molecule gains mass as it gains velocity (but relative to what, in the case of an isolated molecule in space?), "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment.

Consider two isolated molecules, with different velocity vectors. How do we measure their mass? In principle (I'll grant you this bit is well outside the realm of experimental physics) the only way is to measure the gravitational force exerted on a test mass that is stationary with respect to each molecule. And to our (theorietical) astonishment, they both have the same mass.

Back to the ensemble. Even if the said boss is indeed conversant with relativistic physics, he will still need an exceptional grasp of the Gibbs energy of dynamic Voronoi polyhedra if he is going to accept a theoretical proof, and despite advances in chaos theory since I last worked on shortrange order in water, the numbers remain very vague and entirely experimental.

So we have to subtract the calculated relativistic contribution to the ensemble mass from the currently-incalculable density variation.

Once again, I have to ask you to do the calculation, on the assumption that the said boss will understand it. I'll be generous here and let you use RD's (experimental) graph, just this once, to check your answer.  And perhaps you will let us know, as an aside, how many angels can sit on the head of a pin!

Frankly, I think putting a wine bottle in a freezer is a lot easier. But then I'm an experimentalist.

PS - belated good Thanksgiving wishes. Thanks to the Pilgrim Fathers I now have plenty of wine bottles and an empty freezer!

I don't see why you would concede at all. Weight is simply a measure of gravitational force. Does mass really increase with velocity? If you weigh something on a mountain top there will be a very small difference to the weight at sea level. On a different planet things would be much different. 1KG would no longer make sense. I doubt if we can say the mass has increased or decreased in those circumstances. The same could be said for relativistic velocities. It is just harder to move a massive object like a particle at those velocities so it appears to gain mass. Photons have no such problem. I will always side with an experimentalist over a theorist.
 

Offline lightarrow

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Quote from: alancalverd
Except that the water is  hotter. This means that the molecules have more kinetic energy, not that they have gained in mass - you can't have your cake and eat it!
In Newtonian mechanics this is true. In relativistic mechanics its not true. However in the later case the increase is very very small. Too small in fact to be measured in a typical lab. One needs a mass spectrometer to measure such a small increase.
I haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what? And what are you saying about a mass spectrometer? That you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.

--
lightarrow.
« Last Edit: 28/11/2014 20:29:32 by lightarrow »
 

Offline PmbPhy

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Quote from: lightarrow
I haven't understood what you say here. Are you talking of a mass increase of a particle with speed or what?
Yes. I'll restate it to make it clearer. In what follows what I mean by the use of the term mass I mean inertial mass in the Newtonian case and relativistic mass in the relativistic case.

In Newtonian mechanics mass does not increase with speed.
In relativistic mechanics mass does increase with speed.

Simple, yes? :)

Quote from: lightarrow
And what are you saying about a mass spectrometer?
That's one way to measure the relativistic mass of a charged particle.

Quote from: lightarrow
That you could measure a particle's mass increase with speed using that device? If you are saying this, it's incorrect, and not because it would be too small, but because you can't.
That's  incorrect. In fact Wikipedia states that in 1901 Walter Kaufmann used a mass spectrometer to measure the relativistic mass increase of electrons. See http://en.wikipedia.org/wiki/History_of_mass_spectrometry

See also the article in American Journal of Physics about this particular point at
http://gabrielse.physics.harvard.edu/gabrielse/papers/1995/RelativisticMassAJP.pdf

However a mass spectrometer is in essence a cyclotron which can measure a mass increase due to relativistic mass.

The physics principles are the same as they are in a cyclotron. The particle is charged and that makes it move on a circle in a magnetic field. The radius of the circle it moves in. The physics for a cyclotron and a mass spectrometer is in the following webpage in my website: http://home.comcast.net/~peter.m.brown/sr/cyclotron.htm

Quote from: lightarrow
If you are saying this, it's incorrect, and not because it would be too small, but because you can't.
Please provide a proof of this. A derivation whose results show what you assert would be fine. Thanks.
 

Offline alancalverd

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Quote from: alancalverd
Consider two isolated molecules, with different velocity vectors. How do we measure their mass?
Their relativistic mass does not depend on its direction so we only need to measure its mass as a function of speed.
That's why I asked you to consider two particles with different velocity vectors. To make it closer to a practical experiment, think of two identical rocks hurtling at constant speed through space on nonintersecting courses. Their speed relative to each other (a) is variable and different from their fixed speeds (b and c) relative to a third-party frame of reference, which are also different. But their masses, as determined by the gravitational force on a test particle travelling with each (the only way you can determine the mass of a free asteroid), must be constant and identical because there is no universal frame of reference and neither rock is moving with respect to its test mass. Or variable with a. Or fixed but different according to b and c. Please choose, and show your reasoning.   

Quote
To do that one can measure the mass of a moving particle such as a molecule by ionizing it and using a cyclotron to measure its mass by determining it's deflection by a magnetic field and measuring its radius of orbit.

You're an experimental physicist aren't you? If so they why didn't you know this?

I not only knew it, but recently used it in another thread as an example of the experimental proof of relativity!

Quote from: alancalverd
Once again, I have to ask you to do the calculation, ...
I don't see the point. I'm going to assume that "michael clark" is satisfied.
[/quote]

He asked for a proof that would convince his boss. I gave him a simple experiment that showed water in the range below 4C was lighter than water at 4C and this was unique, but you showed that thanks to the relativistic effect it always got heavier as you increased the temperature without bound, whilst RD presented a graph that contradicted that statement, so it seems that some calculation is in order if Mr Clark is not to go away completely confused! 
 

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