[thedoc (OP)]

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?

[alancalverd]

This is only true in the region 0  - 4°C, 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 5°C, 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.   

[PmbPhy]

This is only true in the region 0  - 4°C, 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 5°C, 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.

[evan_au]

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.

[alancalverd]

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

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!

[Roadrunner]

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.

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.

[lightarrow]

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 😊).

--
lightarrow

[lightarrow]

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"?

--
lightarrow

[CliffordK]

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 1°C (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 1°C (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 1°C 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.

[jeffreyH]

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 1°C (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 1°C (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 1°C 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.

[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! 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. 

[PmbPhy]

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?

[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.

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.   

[PmbPhy]

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

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

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.

[alancalverd]

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

[RD]

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 0^oC to 4^oC  ...

... in water below about 4 °C, warmer water sinks whereas when above about 4 °C, warmer water rises ..."

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[PmbPhy]

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

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

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: ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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

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.

[alancalverd]

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!

[PmbPhy]

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!

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
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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
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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 can’t 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.

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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

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.

... "hotter" is not defined for a single molecule - temperature is an ensemble property, but we'll return to that in a moment.

See again ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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.

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

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.

[PmbPhy]

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
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Similarly you might consider that your cup of coffee is more massive when it’s 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 it’s 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.

[lightarrow]

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

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

--
lightarrow

[jeffreyH]

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.

[lightarrow]

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.

[PmbPhy]

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?

And what are you saying about a mass spectrometer?

That's one way to measure the relativistic mass of a charged particle.

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 ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

See also the article in American Journal of Physics about this particular point at
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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: ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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.

[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.   

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!

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 4°C was lighter than water at 4°C 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! 

[PmbPhy]

That's why I asked you to consider two particles with different velocity vectors.

I don't understand what you mean. What do you mean by "That's why I asked...." I don't understand. Exactly why did you ask?

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.   

I don't understand at all what you're leading to or asking me. Please be more clear. Also I don't understand what you mean by "Please choose" What am I choosing?

Also since we're talking about relativistic mass why are you ignoring it in the case of the case of a particle moving through a gravitational field? E.g. see equation 16 in my web page -  ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

The m_p in that equation is the passive gravitational mass of the object and it depends on both the field and the gravitational potential and as such the position of the particle in the gravitational field.

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

Then why were you asking me how to measure it rather than posting it yourself? Unless it didn't come to your mind?

He asked for a proof that would convince his boss. I gave him a simple experiment that showed water in the range below 4°C was lighter than water at 4°C ....

If this

Fill a cup close to the brim with water, put  it in the freezer, and stir it with a thermometer. When it reaches about 5°C, 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.

is true then its not because there's a change in mass but a change in density. It's simply not possible to measure changes in mass that small in situations like this. In fact today there is no experiment of any kind which can show the mass increase for a macroscopic body due to an input/output of heat.

If you have the text Spacetime Physics - Second Edition by Taylor and Wheeler, then turn to page 223 and you'd see that they communicated with Vladimir Braginsky at the University of Moscow about a device to measure the increase in the weight of a tiny heated quartz pellet. They concluded that the technology does not exist to detect it yet. That it surpasses the present limit of technology

He asked for a proof that would convince his boss. I gave him a simple experiment that showed water in the range below 4°C was lighter than water at 4°C 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! 

The reason I gave a relativistic description is because there is no classical mechanism that allows a water to become lighter. It just can't happen. It would mean that the law of the conservation of mass would be violated. It's only violated when relativistic effects are taken into account. You mistook the effects of the change in the density of water with a change in weight and therefore a change in mass.

See:
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I'll post that section of Taylor and Wheeler's text after I scan it in and upload it to my website.

[CliffordK]

He asked for a proof that would convince his boss. I gave him a simple experiment that showed water in the range below 4°C was lighter than water at 4°C and this was unique,

The OP hasn't replied yet, so it is difficult to know the ultimate intentions. 

Yes, most materials have a positive expansion coefficient (decrease in density when heated), except for a few materials in specific circumstances, such as water that has a negative expansion coefficient between 0°C and 4°C.  Of course, that isn't a change in mass, but would be visible as a change in volume in a sealed container.

As far as Einstein E=mc², that would be more difficult to prove.  I derived some of the quantities on paper, but that is far from a rigorous empirical proof, and it would be difficult to weigh a cubic km of water or ice to an accuracy of a few grams.

I'm not sure a mass-spec has high enough of accuracy.  It also requires ions which may be problematic for a liquid water sample, but should still be representative.  Is, say, altering the temperature of a set of ions representative of the change in mass, or are there other confounding variables?

[PmbPhy]

Here is that page from the text Spacetime Physics 2nd Ed that I mentioned
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The OP hasn't replied yet, so it is difficult to know the ultimate intentions.

My impression is that the rarely do anyway.

Of course, that isn't a change in mass, but would be visible as a change in volume in a sealed container.

Yup. It happens every time I make ice cubes. I'm very surprised that one got by an experimental physicist like Alan.

As far as Einstein E=mc², that would be more difficult to prove.

As I indicated above, one does not "prove" things in physics other than those cases when it's a mathematical proof. One forms postulates, makes predictions from those postulates, designs experiments from those predictions and observes what happens in the experiments. If the experiment is consistent with the predicted results then one has more confidence in the theory. Otherwise one has to either abandon the theory or modify it. See

What is Science - ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN
The first paragraph reads

The following statement was originally drafted by the Panel on Public Affairs (POPA) of the American Physical Society, in an attempt to meet the perceived need for a very short statement that would differentiate science from pseudoscience. Am. J. Phys. 67 (,

I derived some of the quantities on paper, but that is far from a rigorous empirical proof, and it would be difficult to weigh a cubic km of water or ice to an accuracy of a few grams.

I'd enjoy reading what you derived. Can you post it here for me or send it to me in PM or e-mail? Thanks. Or please take a look at
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN and let me know if it's the same thing.

I'm not sure a mass-spec has high enough of accuracy.

It depends on the particular mass spectrometer. In any case, as I said above the physics is the same as a cyclotron so if the spectrometer doesn't work merely use a spectrometer.

It also requires ions which may be problematic for a liquid water sample, but should still be representative.

I'd say that a ion of H₂O would be just fine. However it'd be better to find out why he's so concerned with water as opposed to any other substance or element. If we find that out then it'd make things easier.

Is, say, altering the temperature of a set of ions representative of the change in mass, or are there other confounding variables?

At this point it's time to talk to the OP and his boss and get details. If he is still around that is. He might have forgotten all about this. After all he asked the question 4 days ago and we haven't heard even a whisper from either of them since.

[PmbPhy]

I'm not sure a mass-spec has high enough of accuracy.

Note: The reason I mentioned a mass-spectrometer is because it works like a cyclotron which can be used to measure the relativistic mass of a particle or ion. So tell you what we can do. Instead of using a mass-spectrometer, let's use a cyclotron instead. That way we don't have to worry about it working.

Of course Mike's boss can't get his hands on a cyclotron so that's the end of that.   Besides, the physics is too hard to work with when quantities are that small and increases that small too.

[alancalverd]

This is all fascinating stuff, but from the point if view of psychology and philology, not physics!

A benchtop mass spectrometer running at a few keV probably won't show the relativistic change of mass to a detectable level (it's calibrated with known ions - chemistry rather than physics - because it is very difficult to measure static magnetic fields to sufficient accuracy to make an absolute measurement of mass), but a hospital cyclotron certainly will, and we have to allow for it in designing the electronics.

But the real point of argument is the meaning of "heavier". Given that the original questioner didn't know how to demonstrate the effect, I think his enquiry was based on the common knowledge and observation of the anomalous expansion of water rather than the subtle timing corrections of a cyclotron driver - especially as you can't accelerate a water molecule in a cyclotron. Therefore he was talking about bulk density, not molecular mass.   

Is polystyrene heavier than gasoline? To the layman, the answer is clearly no because the "everyday" forms are foam and liquid respectively, but as a liquid or solid, the answer is yes, it is a lot denser. and if you want to be picky, it even has a higher molecular weight. But the molecular weight of gasoline is much greater than that of water, so surely water will float on gasoline? If only it did so, firefighting would be a lot easier!

[PmbPhy]

This is all fascinating stuff, but from the point if view of psychology and philology, not physics!

I can't see how that could possibly true at all. All I've done is to directly answer the OPs question and tried to do it in manner to answer what his boss is probably seeking the answer too rather than what he literally asked for proof of.

But the real point of argument is the meaning of "heavier".

I can't possibly see how that's a point of disagreement since most non-physicists confuse weight with mass. It's very clear to me that he wants to know which of two initially identical cups of water will have a greater mass will have a greater mass when one of them is heated, i.e. energy is added to the water. Most people I know are happy with thought experiments so I'd wager that Michael doesn't actually want to go out and do an experiment since he mentioned nothing about experiments. When he said "proof" it seemed to me that he wants to present a solid argument to present to him which will convince him that he's right.

Given that the original questioner didn't know how to demonstrate the effect, I think his enquiry was based on the common knowledge and observation of the anomalous expansion of water ...

I see absolutely no reason to assume that'd be the case. That is a major jump in my opinion.

..rather than the subtle timing corrections of a cyclotron driver - especially as you can't accelerate a water molecule in a cyclotron.

That's not true at all. And what's a "cyclotron driver" anyway and how is it related to this topic? There is absolutely no reason why a molecule of water, i.e. an H₂O ion couldn't be accelerated in a cyclotron. The following is close to what I'm referring to:
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From my Google search I came up with this - ion cyclotron resonance mass spectrometry which appears to be what can be used in principle to measure the mass of water ion.

In any case, why are we so worried about experiments. Michael said nothing about experiments or that the answer had to be able to be done in practice. For all we know, and I know that if it were I then it's what I'd be looking for, I'd want to know what's going on in principle.


re - especially as you can't accelerate a water molecule in a cyclotron. - There is no basis for such an assumption. On what are you basing that assertion on?

Therefore he was talking about bulk density, not molecular mass.   

You don't know that at all. He specifically asked which was heavier, hot water or cold water. And as I said for most people, and nearly universally all layman, use weight synonymously for mass.


Sorry Alan but to me you're just grasping at straws now.

[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?

And what are you saying about a mass spectrometer?

That's one way to measure the relativistic mass of a charged particle.

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 ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

See also the article in American Journal of Physics about this particular point at
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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: ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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.

In order to analyze relativistic dynamic experiments, what we need is:

E² = (c*|p|)² + (m*c²)²     (1)
v = c²*p/E     (2)

(p and v are momentum and velocity vectors, "| |" means modulus of the vector).
From these two equations we can deduce other relativistic equations, for example:

E = mc²*γ
p = M*v*γ

Where γ = 1/sqrt{1 - (|v|/c)²}.

We also need the law:

F = dp/dt     (3)

which is the definition of force in relativistic dynamics.

Let’s analyze the motion of a charged particle in a magnetic field.
We know that (Lorentz force):

F = q*v x B     (4)

Where “x” means vectorial product.

So F is always orthogonal to v, and orthogonal to p too, according to eq. (2). It follows that the modulus of p is constant (and according to eq. (1) even E is constant):

d|p|²/dt = 2 p.(dp/dt) = 2 p.F = 0.

(the dot between two vectorial factors means scalar product).

Defining the angular speed as: ω = |v|/r, according to kinematics we have:

|dp/dt| =  ω*|p|

(This equation is valid in galileian as well as in relativistic kinematics).

For the sake of simplicity, let’s assume v and p both orthogonal to the magnetic field B.
Then the trajectory is a circle, traveled of uniform motion and which lays in a plane orthogonal to B.
So, using equations (2), (3), (4) and the expression of the (modulus of) Lorentz force on a charged particle:

|F| = q*|v|*|B|

we easily deduce :

ω*|p| = q*|v|*|B|     (5)

|p|/r = q*|B| → |p| = q*|B|*r    (6)

and it’s this last equation that we can use for a mass spectrometer, measuring r and knowing |B|, to compute |p| and so to find the mass m according to eq. (1), knowing the total energy E. The last can be found using also:

E = E_k + m*c²     (7)

where the kinetic energy E_k is set by the potential difference V applied to the charge q: E_k = q*V.

Substituting in eq. (7) and then in eq. (1):

(q*V + m*c²)² = (c*|p|)² + (m*c2)²     

from which you can find the  invariant mass m of the charged particle.

No need of relativistic mass.

N.B. (Most of what written up is not mine, it comes from another source, but it's rather simple to understand.
The source is an italian thread started from prof. Elio Fabri:
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN).

But if you intended that the mass spectrometer allows you to find the momentum |p| and so the total energy E and you identify the last with relativistic mass (multiplied c²), then we again come back to what I have said before several times in various threads, even answering to you, that is that relativistic mass is nothing else than another name for total energy E.

Edit: coloured in blue eq. (6) - 01/12/2014

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[jeffreyH]

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?

And what are you saying about a mass spectrometer?

That's one way to measure the relativistic mass of a charged particle.

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 ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

See also the article in American Journal of Physics about this particular point at
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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: ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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.

In order to analyze relativistic dynamic experiments, what we need is:

E² = (c*|p|)² + (m*c²)²     (1)
v = c²*p/E     (2)

(p and v are momentum and velocity vectors, "| |" means modulus of the vector).
From these two equations we can deduce other relativistic equations, for example:

E = mc²*γ
p = M*v*γ

Where γ = 1/sqrt{1 - (|v|/c)²}.

We also need the law:

F = dp/dt     (3)

which is the definition of force in relativistic dynamics.

Let’s analyze the motion of a charged particle in a magnetic field.
We know that (Lorentz force):

F = q*v x B     (4)

Where “x” means vectorial product.

So F is always orthogonal to v, and orthogonal to p too, according to eq. (2). It follows that the modulus of p is constant (and according to eq. (1) even E is constant):

d|p|²/dt = 2 p.(dp/dt) = 2 p.F = 0.

(the dot between two vectorial factors means scalar product).

Defining the angular speed as: ω = |v|/r, according to kinematics we have:

|dp/dt| =  ω*|p|

(This equation is valid in galileian as well as in relativistic kinematics).

For the sake of simplicity, let’s assume v and p both orthogonal to the magnetic field B.
Then the trajectory is a circle, traveled of uniform motion and which lays in a plane orthogonal to B.
So, using equations (2), (3), (4) and the expression of the (modulus of) Lorentz force on a charged particle:

|F| = q*|v|*|B|

we easily deduce :

ω*|p| = q*|v|*|B|     (5)

|p|/r = q*|B| → |p| = q*|B|*r    (6)

and it’s this last equation that we can use for a mass spectrometer, measuring r and knowing |B|, to compute |p| and so to find the mass m according to eq. (1), knowing the total energy E. The last can be found using also:

E = E_k + m*c²     (7)

where the kinetic energy E_k is set by the potential difference V applied to the charge q: E_k = q*V.

Substituting in eq. (7) and then in eq. (1):

(q*V + m*c²)² = (c*|p|)² + (m*c2)²     

from which you can find the  invariant mass m of the charged particle.

No need of relativistic mass.

N.B. (Most of what written up is not mine, it comes from another source, but it's rather simple to understand.
The source is an italian thread started from prof. Elio Fabri:
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN).

But if you intended that the mass spectrometer allows you to find the momentum |p| and so the total energy E and you identify the last with relativistic mass (multiplied c²), then we again come back to what I have said before several times in various threads, even answering to you, that is that relativistic mass is nothing else than another name for total energy E.

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lightarrow

I am so glad that you posted this. I don't think many will appreciate the subtleties.

[alancalverd]

Yup. It happens every time I make ice cubes. I'm very surprised that one got by an experimental physicist like Alan.

The OP concerned water, not ice. Expansion on solidification is not unusual - printers' "hot type" and several other materials do it - but the expansion of liquid water over a significant range of cooling is remarkable and very important. 

And what's a "cyclotron driver" anyway and how is it related to this topic?

It's the electronic gubbins that turns the cyclotron from a drawing into a working machine. But don't worry your pretty head about that nasty engineering stuff - you'll get you hands dirty! 

There is absolutely no reason why a molecule of water, i.e. an H2O ion couldn't be accelerated in a cyclotron.

A molecule is not an ion. Which seems a good reason to me.

[jeffreyH]

Yup. It happens every time I make ice cubes. I'm very surprised that one got by an experimental physicist like Alan.

The OP concerned water, not ice. Expansion on solidification is not unusual - printers' "hot type" and several other materials do it - but the expansion of liquid water over a significant range of cooling is remarkable and very important. 

And what's a "cyclotron driver" anyway and how is it related to this topic?

It's the electronic gubbins that turns the cyclotron from a drawing into a working machine. But don't worry your pretty head about that nasty engineering stuff - you'll get you hands dirty! 

There is absolutely no reason why a molecule of water, i.e. an H2O ion couldn't be accelerated in a cyclotron.

A molecule is not an ion. Which seems a good reason to me.

A molecule yes but look at the hydrogen bonding.

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[PmbPhy]

The OP concerned water, not ice.

I didn't say he was. It was merely a thought on expansion of cold water.

But don't worry your pretty head ...

So this is what you resort to when you loose an argument, huh?

I know a great deal about engineering since I worked in the field for years before becoming a physicist.

A molecule is not an ion. Which seems a good reason to me.

An ion is an atom or a molecule for which the number protons is not the same as the number of electrons. That's how atoms and molecules can be accelerated in a cyclotron. And you claim to be an experimental physicist and you didn't know this simple fact that all physicists and chemists know?

You should learn the basics of physics before insulting and being rude to others. Start with these:

...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

An ion (/ˈaɪən, -ɒn/)[1] is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving the atom or molecule a net positive or negative electrical charge.

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Ion cyclotron resonance is a phenomenon related to the movement of ions in a magnetic field. It is used for accelerating ions in a cyclotron, and for measuring the masses of an ionized analyte in mass spectrometry, particularly with Fourier transform ion cyclotron resonance mass spectrometers. It can also be used to follow the kinetics of chemical reactions in a dilute gas mixture, provided these involve charged species.

See also - ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

: an atom or group of atoms that has a positive or negative electric charge from losing or gaining one or more electrons

See the list of molecules in that page that they use as examples of ions.

[PmbPhy]

No need of relativistic mass.

So what? Do you realized just how many concepts there are in physics that aren't "needed"? In fact Hertz wrote an entire mechanics text without using the concept of force. I could write an entire textbook on relativity without using the concept of energy either. Let me show you how easy it'd be to derive the cyclone formula by using relativistic mass. Let B = Bk where B is a constant. Let v start out in the xy-plane. Then

F = qvxB

means that the force is perpendicular to the velocity which means the charge moves in a circle and the force is perpendicular to the velocity. As such F = ma. If you're not aware that this is true then see
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

F = ma = mv²/r = qvB

or

mv = qrB = p

which is the cyclotron relation, i.e.

mv = qrB

Which requires a great deal less work that what you did.

[PmbPhy]

A molecule yes but look at the hydrogen bonding.

You're wrong too, Jeff. A molecule which has a different amount of electrons than protons is an ion. Same holds true with atoms.

[CliffordK]

As far as heat and relativistic mass, if it could be demonstrated for any substance, water, hydrogen, helium, radon, lead, etc, then it could be considered valid for all substances. 

While H₂O is not an ion, water is always a mix of ions, H₂O, OH^–, H⁺, and H₃O⁺.  With acid/base chemistry, it is relatively easy to force one or another ion to be more prevalent, at last in liquid water form.  Gases may tend to be more neutral.  There is no reason why water couldn't be used as a source of ions for analysis.  However, there may be better choices of materials for analysis.

[alancalverd]

A molecule is by definition electrically neutral and thus cannot be accelerated by a cyclotron.

An ion may have the same nuclides, almost the same mass and even most of the shape of a molecule but it is distinguished by having a charge, and thus quite different behaviour in an electric or magnetic field.  Which is why it has a different name, and why we ionise molecules in a mass spectrometer.

It is worth reconsidering the Wikipedia definitions. If I have an ion with 6 protons and 7 electrons, is it carbon plus an electron or nitrogen minus a proton? If we incorporate the nuclide into a fairly simple compound, the difference is between an inert plastic and an explosive, so we need to be a bit careful with our definitions here. 

[PmbPhy]

A molecule is by definition electrically neutral and thus cannot be accelerated by a cyclotron.

You're posting nonsense now. I said it was an ion of a H₂O, not a water "molecule." I also made it clear by posting definitions that an ion is a molecule that is not neutral (you don't appear to be reading the definitions that I've posted to help you grasp the nomenclature) which means that you start off with a molecule and then add or subtract electrons to create the ion. I was using ions because they can be accelerated in a cyclotron.

When I kept referring to ions of a H₂O, and you kept changing it to a water molecule you introduced a logical fallacy known as a straw argument or straw man.

An ion of  H₂O (i.e. a water ion) is not neutral and therefore can be accelerated. Since the mass is very close to that of a neutral molecule it will suffice for the purpose I was arguing, i.e. to measure the mass of a particle of water (i.e. a water molecule).

Since you've turned to using logical fallacies such as this then I can't see any reason to continue making an attempt to reason with you. Clearly you don't understand the nomenclature too. In the physics/chemistry community when speaking of ions one uses the term "ion" (or cation, anion ect) and THEN name of the related molecule. See the tables in ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN  so one says "ion of water" etc.

You started off in this thread confusing density with weight so you're just too illogical for me.

[PmbPhy]

While H₂O is not an ion, ...

Now you're making the same mistake alan was. I never said a molecule of water/H₂O. I kept saying and ion of H₂O. People really need to read more carefully before criticizing what they're reading.

[PmbPhy]

A molecule yes...

It appears that you're like others and didn't read what I posted carefully. I only spoke of H2O ions. I never spoke of H2O molecules.

[alancalverd]

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 are right - I missed "by ionizing it". Mea culpa - unless that was a late edit, in which case you are a scoundrel, sir!

Pistols at dawn? Should give me a 5 hour advantage.

And anyway, if you ionize a water molecule by removing a proton, you will have changed its mass, its chemistry, and its bulk physics, significantly. So there! []

[lightarrow]

...
|p| = q*|B|*r
...   

I am so glad that you posted this. I don't think many will appreciate the subtleties.

That's true. Prof. Elio Fabri has taught Theoretical Physics at the univeristy of Pisa. He specifically taught courses of Quantum Mechanics, Special and General Relativity, Quantum Field Theory, Astrophysics, Classical Mechanics and Electrodynamics, among other subjects.

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[lightarrow]

A benchtop mass spectrometer running at a few keV probably won't show the relativistic change of mass to a detectable level
...
but a hospital cyclotron certainly will, 

No, never. It will only show the relativistic change of momentum or of energy E.

*If then you define* relativistic mass R.M. = E/c² then of course it would show relativistic increase of that, but it's just the energy E (apart a constant multiplied to it) with another name!

Neither a mass spectrometer nor a cyclotron nor *any other device* will ever show a relativistic change of mass.

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[lightarrow]

No need of relativistic mass.

So what? Do you realized just how many concepts there are in physics that aren't "needed"? In fact Hertz wrote an entire mechanics text without using the concept of force. I could write an entire textbook on relativity without using the concept of energy either. Let me show you how easy it'd be to derive the cyclone formula by using relativistic mass. Let B = Bk where B is a constant. Let v start out in the xy-plane. Then

F = qvxB

means that the force is perpendicular to the velocity which means the charge moves in a circle and the force is perpendicular to the velocity. As such F = ma. If you're not aware that this is true then see
...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

F = ma = mv²/r = qvB

or

mv = qrB = p

which is the cyclotron relation, i.e.

mv = qrB

Which requires a great deal less work that what you did.

Why didn't you write the equation:

F = ma = mv²/r = qvB

with F, a, v, B in bold carachter? It means that it's not valid vectorially? (Of course it's not  []) And what usefulness it has an equation involving vectors that it's not valid vectorially? It can be useful just in very special cases like this one, just because in my example I used a particle's initial velocity v orthogonal to B (and you correctly took the same example because you knows well that F = ma it's not true in general  []). Try to make the computations with any v!  []

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[PmbPhy]

lightarrow - I don't know why you're trying so hard to make this a debate about relativistic mass but I'm really not interested and I won't discuss it.

[lightarrow]

lightarrow - I don't know why you're trying so hard to make this a debate about relativistic mass but I'm really not interested and I won't discuss it.

In your first post of the thread "A poll on relativistic mass" you asked yourself why I believe relativistic mass is merely another name for energy, now you know  [].

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[PmbPhy]

Why didn't you write the equation:

F = ma = mv²/r = qvB

with F, a, v, B in bold carachter?

Because I had no use for it. I wrote down the magnitudes involved rather than the vectors since it was much easier to do so and its exactly correct. One should always make things easier when one can. Didn't you know this about using the equivalent scalar equations from the associated vector equations when possible? If not then I find that surprising. You saw how easy it was doing so. It'd take more work not to. Besides, that's how most physicist derive it, i.e. using the scalar equation and not the vector equation. I take it that you didn't know that either?

Also, you didn't really say why you believe that relativistic mass is merely another name for energy. All you really did is make a baseless assertion that it is merely another name. However it is in fact not merely a different name for it. There are many reasons why but the real and most obvious reason is that explained in the journal article On The Meaning of E = mc² by Mendel Sachs, Int. J. Theo. Phys., 8 (1973) who writes

... inertia and energy refer to conceptually different features of matter. On the one hand, inertia per se relates to the resistance to the change of state of motion of a body, as would be caused, in 'particle physics,' by an external force acting on the body (...).
   On the other hand, energy per se is defined in terms of the solutions of the conservation laws.....
Thus we see that energy and mass are conceptually different (though complimentary) features of matter, according to relativity theory.

Several relativists have made this argument, including me. Just because two quantities are proportional it doesn't mean that they are the same thing. E.g. you can't legitimately claim that since the energy is related to the frequency of a photon by E = hf that energy and frequency are the same thing.

Besides, E = mc² does not work under all situations. E.g. if you have a rod which is at rest lying along the x-axis in the inertial frame S and there is  force on each end of the rod of equal magnitude but opposite directions so as to stress it then it will be no acceleration of the rod but it will have a finite energy. When you transform to S' which is in standard configuration with S and moving in the +x direction then E = mc² will no longer be true.

Also, if you are in a non-inertial frame E = mc² will also not be true. The energy will be proportional to the time component of the momentum 1-form. However relativistic mass will still be the time component of the 4-momentum.

I didn't want to talk about your mistakes on this subject but since you took this off the thread where I was only looking for opinion you changed the purpose of it.

Nothing personal my friend.